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Chapter 8

Dynamical Systems

JORGE REBAZA

In this chapter we give an introduction to the very important subject of dynam-ical systems, the study of qualitative behavior and computation of solutions ofnonlinear systems of the form

x = f(x), f : Rn → R

n. (8.1)

We present some of the most relevant theoretical results in this area, as well asseveral examples to better illustrate the idea being introduced. We also takecare of introducing some numerical techniques necessary for the computation ofspecial solutions. Some particular mathematical models of dynamical systemsare studied in detail, and give us a chance to highlight and apply the theory andtechniques introduced.

In most mathematical theory, dealing with linear problems is by far simpler thanstudying nonlinear ones. Results on linear dynamical systems are well estab-lished, and are essential to understand and develop the theory for their nonlinearcounterpart. Thus, our starting point will be focusing on the simpler theory oflinear dynamical systems and the qualitative behavior of their solutions.

The study of nonlinear dynamical systems is usually divided into local and globaltheory. The main ingredient for local theory will be the linearization of the sys-tem around equilibrium points or periodic solutions, and then we will study theresulting linear systems to understand the local behavior of the correspondingnonlinear system around those special solutions. We will also consider some as-pects of global theory, by studying some special solutions such as homoclinic andheteroclinic orbits, and we will give an elementary introduction to bifurcations,

391

392 CHAPTER 8. DYNAMICAL SYSTEMS

which represent radical changes in qualitative behavior of solutions for smallchanges in the problem parameters. Finally, we will give a brief introduction tochaos, characterized by unpredictable behavior of solutions.

8.1 Linear Dynamical Systems

Most problems in real-world applications are nonlinear and our main focus inthis chapter is on nonlinear dynamical systems. But as explained above, the localtheory of nonlinear systems can be studied through linearization; therefore it isnot only very illustrative but especially essential to first study and understandlinear systems.

We have already seen in Section 7.2 (see (7.57) and (7.58)) that for an n × nmatrix A, the solution to the initial value problem (IVP)

x = Ax, x(0) = x0 (8.2)

is given by

x(t) = eAtx0, t ∈ R, (8.3)

where the matrix exponential is:

eAt =∞∑

k=0

Aktk

k!. (8.4)

Firstly, from (8.3), we can deduce that there is no unpredictability in the solutionof a linear system, since such solution is well determined and known for all t ∈ R,and therefore we do not expect to have chaotic solutions. Secondly, we observethat the dynamics of the solution is contained in the matrix exponential eAt, ormore precisely, it is the eigenvalues of the matrix A that will tell us about thestability properties of the solutions.

In this context, the concept of similarity of matrices becomes particularly im-portant in the study of linear dynamical systems. The following theorem saysthat if two matrices are similar, so are their corresponding matrix exponentials.

Theorem 8.1 Let A and B be two n× n matrices. Then

8.1. LINEAR DYNAMICAL SYSTEMS 393

A = PBP−1 implies eAt = PeBtP−1. (8.5)

In particular, if A is similar to a diagonal matrix D, we have that

A = PDP−1 implies eAt = P

eλ1

. . .

eλn

P−1, (8.6)

where the diagonal entries of D : λ1, . . . , λn, are the eigenvalues of A.

Proof. First observe that if A = PBP−1, then Ak = PBkP−1, for anynonnegative integer k. Then

eAt =∞∑

k=0

Aktk

k! = limn→∞

n∑

k=0

Aktk

k! = limn→∞

n∑

k=0

PBkP−1 tk

k!

= P

(

limn→∞

n∑

k=0

Bktk

k!

)

P−1 = P

(

∞∑

k=0

Bktk

k!

)

P−1

= PeBtP−1.

If B = D is diagonal, so is eBt, as in (8.6).

�

Some terminology. Before we start studying the qualitative behavior of solu-tions we need to introduce some notation and terminology, with linear systemsin mind. Some of this notation and terminology will later be extended to thenonlinear case.

Let E be an open set of Rn. We say that the function φ : R × E → E defined

by

φ(t, x) = eAtx (8.7)

defines a dynamical system on E ⊆ Rn.

If we relate this definition with the system x = Ax, and its solution x(t) = eAtx0,we observe that the dynamical system defined by φ(t, x) is a description of howa given point or state x ∈ E moves with respect to time; that is, how a solutionx = x(t) of x = Ax evolves with time.


The solution of x = Ax defines a motion that can be described geometricallyby drawing the solution curves in the R

n space, which in this context is calledthe phase space (phase plane, in two dimensions).

The phase portrait of a dynamical system is the set of all solution curves on thephase space.

Clearly, the origin x = 0 satisfies Ax = 0, that is, x = Ax = 0, for all t ∈ R. Thismeans the origin does not change with time and we say that it is an equilibriumpoint of x = Ax.

8.1.1 Dynamics in two dimensions

On our way to understand n-dimensional linear systems, we first study thevery illustrative and helpful two-dimensional case. The focus is on classifyingthe linear dynamical systems according to the eigenvalues of the correspondingmatrix A. We know that in two dimensions the eigenvalues of a matrix will beeither real and different, real and repeated, or complex conjugate, and thereforethese cases will determine the various qualitative behaviors of the solutions.

Thus, we start considering the matrices B below and their corresponding matrixexponentials (see also Exercises 8.10 and 8.11):

B =

[

λ1 00 λ2

]

=⇒ eBt =

[

eλ1t 00 eλ2t

]

,

B =

[

λ 10 λ

]

=⇒ eBt =

[

eλt teλt

0 eλt

]

= eλt[

1 t0 1

]

,

B =

[

a −bb a

]

=⇒ eBt = eat[

cos bt − sin btsin bt cos bt

]

.

(8.8)

Remark 8.2 The important fact is that given an arbitrary 2 × 2 matrix A, itcan be shown that it is similar to one of the matrices B in (8.8). This meansthat since the solution of (8.2) is given in terms of a matrix exponential, andrecalling Theorem 8.1, we can restrict ourselves to the cases in (8.8) to coverall possible solutions of (8.2) for n = 2.


In the two-dimensional case we can take real advantage of graphical illustrationsto facilitate the study of a dynamical system. This will hep us understand thebehavior of solutions, especially as t increases to infinity, and to classify theorigin (equilibrium point of the system) according to the asymptotic behaviorof the solutions.

As remarked before, the eigenvalues of the matrix A determine the qualitativebehavior of solutions around the origin. In general, presence of at least oneeigenvalue with positive real part will cause the solutions to get farther andfarther from the origin as t → ∞ In this case, the origin is called unstable.If all eigenvalues have zero real part, then the solutions will stay around theorigin, but without approaching it as t → ∞. The origin is called stable. Ifall eigenvalues have negative real part, the solutions will approach the originasymptotically (as t→ ∞), and the origin is called asymptotically stable.

We want to look at all the possible phase portraits of a two-dimensional dynam-ical system x = Bx. That is, following Remark 8.2, we consider the effect of thematrices B of (8.8) in the solution x(t) = eBtx0.

Case 1: Eigenvalues with opposite sign.

B =

[

λ1 00 λ2

]

, with λ1 < 0 < λ2. The origin is unstable and it is called saddle

point.

Case 2: Eigenvalues with equal sign.

B =

[

λ1 00 λ2

]

, with λ1 ≤ λ2 < 0, or B =

[

λ 10 λ

]

, with λ < 0. The origin is

called asymptotically stable node.

Case 3: Complex conjugate eigenvalues.

For λ = a±b i , we have B =

[

a −bb a

]

, a < 0. The origin is called asymptotically

stable focus.

Case 4: Pure imaginary eigenvalues.

For λ = ±b i , we have B =

[

0 −bb 0

]

. The origin is called stable center.


Remark 8.3

1. For cases 2 and 3, by considering positive λ1,2 and a > 0 we get the phaseportraits for the corresponding unstable origin.

2. For a general 2 × 2 matrix A, the phase portrait will be equivalent to oneof the four cases above, obtained by a linear transformation of coordinates(similarity transformation).

We illustrate all these cases in the examples below.

Example 8.1.1 (Saddle) Consider the system

x1 = −x1 − 3x2

x2 = 2x2,x0 = [c1 c2]

T . (8.9)

The eigenvalues of A =

[

−1 −30 2

]

are λ1 = −1, λ2 = 2. We inmediately know

that this corresponds to case 1, and thus, the origin is an unstable saddle point.The corresponding eigenvectors v1 = [1 0]T , v2 = [−1 1]T will help determinethe shape and direction of the solution curves in the phase portrait. We keep inmind that v1 is associated to the stable eigenvalue λ1 and v2 is associated to theunstable eigenvalue λ2. Now let us use a similarity transformation to find the

solution x(t). Let P = [v1 v2] =

[

1 −10 1

]

. Then, P−1AP = B =

[

−1 00 2

]

.

Thus, the solution to (8.9) is

x(t) = eAtx0 = P eBtP−1x0 = P

[

e−t 00 e2t

]

P−1x0, or,

x1(t) = c1e−t + c2(e

−t − e2t)x2(t) = c2e

2t,

We also observe that under the transformation u = P−1x, the uncoupled systemis

u1 = −u1

u2 = 2u2,u0 = [k1 k2]

T , (8.10)

with solution u1(t) = k1e−t, u2(t) = k2e

2t. By the similarity transformationA = PBP−1, systems (8.9) and (8.10), and their corresponding solutions, areequivalent. Observe in Figure 8.1 that the phase portraits are equivalent, with


−1 −0.5 0 0.5 1−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8(a)

−2 −1 0 1 2−1.5

−1

−0.5

0

0.5

1

1.5(b)

Figure 8.1: Saddle: (a) Solutions of (8.10). (b) Solutions of (8.9)

the main difference that in (a) the directions are determined by the usual xand y axis (or more properly, by the canonical vectors e1, e2), while in (b) thedirections are determined by the eigenvectors v1 (which coincides with the xaxis) and v2.

We should remark that the above example shows that the similarity transforma-tion is in general a change of coordinates: the solutions in (a) are in the usual xycoordinates, while the solutions in (b) are in the coordinates of the eigenvectorsv1 and v2.

Note: We could have solved system (8.9) by simply solving the second differ-ential equation for x2, and then substituting this into the first equation to solvefor x1, but here we want to illustrate the role of similarity transformations.

Example 8.1.2 (Node) Consider the system

x1 = −5x1 − x2

x2 = −x1 − 5x2,x0 = [c1 c2]

T . (8.11)


[

−5 −1−1 −5

]

are λ1 = −6, λ2 = −4. This corresponds

to case 2, and hence, the origin is an (asymptotically) stable node. The corre-sponding eigenvectors are v1 = [1 1]T , v2 = [1 − 1]T . Let us use a similarity


−2 −1 0 1 2−1.5

−1

−0.5

0

0.5

1

1.5(a)

−2 −1 0 1 2−2

−1.5

−1

−0.5

0

0.5

1

1.5

2(b)

Figure 8.2: Node: (a) Solutions of (8.12). (b) Solutions of (8.11)

transformation to find the solution x(t). Let P = [v1 v2] =

[

1 11 −1

]

. Then,

P−1AP = B =

[

−6 00 −4

]

. Thus, the solution to (8.11) is


[

e−6t 00 e−4t

]

P−1x0, or,

x1(t) = 12 (c1 + c2)e

−6t + 12 (c1 − c2)e

−4t

x2(t) = 12 (c1 + c2)e

−6t + 12 (c2 − c1)e

−4t.

As before, under the transformation u = P−1x, the uncoupled system is

u1 = −6u1

u2 = −4u2,u0 = [k1 k2]

T , (8.12)

with solution u1(t) = k1e−6t, u2(t) = k2e

−4t. By similarity, systems (8.11)and (8.12), and their corresponding solutions, are equivalent. Also, as in theprevious example, observe in Figure 8.2 that the phase portraits are equivalent.

Example 8.1.3 (Focus) Consider the system

x1 = −6x1 − 5x2

x2 = 10x1 + 4x2,x0 = [c1 c2]

T . (8.13)


−2 −1 0 1 2 3

−4

−3

−2

−1

0

1

2

3

4

5

6(a)

−4 −2 0 2

−4

−2

0

2

4

6

8

(b)

Figure 8.3: Focus: (a) Solutions of (8.14). (b) Solutions of (8.13)


[

−6 −510 4

]

are λ1,2 = −1± 5i. This corresponds to case

3, that is, the origin is an asymptotically stable focus. In fact, the solutions willspiral into the origin. The corresponding eigenvectors are v1,2 = [1 − 1]T ±i [0 − 1]T . To find the solution x(t), let P =

[

0 1−1 −1

]

. Then, P−1AP =

B =

[

−1 −55 −1

]

. Thus, the solution to (8.13) is

x(t) = eAtx0 = P eBtP−1x0 = P e−t[

cos 5t − sin 5tsin 5t cos 5t

]

P−1x0, or,

x1(t) = e−t [c1 cos 5t− (c1 + c2) sin 5t ]

x2(t) = e−t [c2 cos 5t+ (2c1 + c2) sin 5t ].

As before, under the transformation u = P−1x, the system is

u1 = −u1 − 5u2

u2 = 5u1 − u2,u0 = [k1 k2]

T , (8.14)

with solution u1(t) = e−t[k1 cos 5t− k2 sin 5t], u2(t) = e−t[k1 sin 5t+ k2 cos 5t].By similarity,systems (8.13) and (8.14), and their corresponding solutions, areequivalent. Also, as in the previous example, observe in Figure 8.3 that thephase portraits are equivalent. Also observe that though the solutions are givenin terms of sines and cosines, the exponential part e−t causes the solutions tospiral towards the origin.


Example 8.1.4 (Center) Consider the system

x1 = 2x1 − 2x2

x2 = 4x1 − 2x2,x0 = [c1 c2]

T . (8.15)


[

2 −24 −2

]

are λ1,2 = ±2i. This corresponds to case 4,

that is, the origin is a stable center. The corresponding eigenvectors are v1,2 =

[1 1]T ± i [0 − 1]T . Now let P =

[

1 01 −1

]

. Then, P−1AP = B =

[

0 2−2 0

]

.

The solution to (8.15) is


[

cos 2t sin 2t− sin 2t cos 2t

]

P−1x0, or,

x1(t) = c1 cos 2t+ (c1 − c2) sin 2t

x2(t) = c2 cos 2t+ (2c1 − c2) sin 2t.

Under the transformation u = P−1x, the uncoupled system is

u1 = 2u2

u2 = −2u1u0 = [k1 k2]

T , (8.16)

with solution u1(t) = k1 cos 2t + k2 sin 2t, u2(t) = −k1 sin 2t + k2 cos 2t. Sys-tems (8.15) and (8.16), and their corresponding solutions, are equivalent. Also,as in the previous example, observe in Figure 8.4 that the phase portraits areequivalent.

Remark 8.4 Observe in Figure 8.4 that the orientation of the trajectories hasbeen reversed. This is because det(P ) < 0. If we had chosen instead P =[

0 1−1 1

]

, then det(P ) > 0, and the orientation would have been preserved. In

general if the complex eigenvector is w = u+ iv, one chooses P = [v u] so thatorientation is preserved. See Example 8.1.3.

Through the examples above, we have covered all possible dynamical systemsin the 2-dimensional case. Although the very shapes of the solution curvesmay vary a little for similar systems, the qualitative behavior around the origin


−1 −0.5 0 0.5 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1(a)

−1.5 −1 −0.5 0 0.5 1 1.5−2

−1.5

−1

−0.5

0

0.5

1

1.5

2(b)

Figure 8.4: Center: (a) Solutions of (8.16). (b) Solutions of (8.15)

remains the same. The central point is that the coefficient matrix A, throughits eigenvalues and eigenvectors, fully determines the solution. The sign of thereal part of the eigenvalues indicates whether the solution is stable or unstable.In the examples discussed, we have considered only stable cases. The unstablecases are identical to the stable cases, but with solutions moving away from theorigin, that is, the direction of the arrows would be reversed.

The associated eigenvectors form a new coordinate system (see e.g. the lineson part (b) of the figures above) through the similarity transformation A =P B P−1, and hence help determine the general shape of the solutions. Forillustration, in all figures above we have plotted first the solutions of the corre-sponding canonical form and then the solutions of the general case in the givenexample.

Degenerate equilibrium point. In the case det A = 0, that is, when oneor both of the eigenvalues of A are zero, then the origin is called degenerateequilibrium point. In such a case, the origin is not the only equilibrium of thesystem. Consider for example the system

x1 = x2

x2 = 0,x(0) =

[

c1c2

]

.

The corresponding matrix A =

[

0 10 0

]

has eigenvalues λ1 = λ2 = 0, and all

the points of the form (x1, 0) are equilibrium points. The solution to the system


x1

x2

Figure 8.5: Degenerate equilibrium point

is

x1(t) = c2t+ c1, x2(t) = c2.

Thus, for c2 > 0, we have x2 > 0, and x1 increases as t increases; similarly, forc2 < 0, we have x2 < 0, and x1 decreases as t increases. The phase portrait isshown in Figure 8.5. See also Exercise 8.14.

8.1.2 Trace-determinant analysis

Here we introduce a very useful result concerning the stability of the origin ofa system x = Ax, for the case when det(A) 6= 0. This technique for determiningthe qualitative behavior of solutions around the origin becomes especially usefulwhen we later consider local study of two-dimensional nonlinear systems aroundequilibrium points and the system itself depends on some parameters. In mostsuch systems, it becomes difficult to find an explicit expression for the eigen-values in terms of the problem parameters, so that a good alternative is to findinstead explicit expressions for the trace and the determinant of the associatedmatrix and then arrive to conclusions about the stability of the origin.

From Theorem 2.59 we know that if det(A) 6= 0, then the only solution of Ax = 0is x = 0, that is, the origin is the only equilibrium of the system x = Ax.


Theorem 8.5 Let A be a 2×2 matrix and denote D = detA, T = traceA, andconsider the system

x = Ax. (8.17)

Then,

(a) If D < 0, then the origin is a saddle point of (8.17).

(b) If D > 0 and T 2 − 4D ≥ 0, then the origin is a stable node of (8.17) ifT < 0 and unstable node if T > 0.

(c) If D > 0 and T 2 − 4D < 0, then the origin is a stable focus of (8.17) ifT < 0 and unstable focus if T > 0.

(d) If D > 0 and T = 0, then the origin is a center of (8.17).

Proof. Let A =

[

a bc d

]

. Then,

det(A− λI) = λ2 − (a+ d)λ+ (ad− bc).

Then, the eigenvalues of A are

λ =T ±

√T 2 − 4D

2.

(a) If D < 0, then√T 2 − 4D > T and therefore A has two real eigenvalues of

opposite sign. This implies the origin is a saddle of (8.17).

(b) If D > 0 and T 2 − 4D ≥ 0, then√T 2 − 4D < T and therefore A has two

real eigenvalues of the same sign as T . This implies the origin is a node; stableif T < 0 and unstable if T > 0.

(c) If D > 0 and T 2−4D < 0 then A has two complex conjugate eigenvalues (aslong as T 6= 0). This implies the origin is a focus; stable if T < 0 and unstableif T > 0.

(d) If D > 0 and T = 0, then A has two pure imaginary complex conjugateeigenvalues. This implies the origin is a center.

�

We can summarize the results of Theorem 8.5 in a diagram representing regionsof different qualitative structure of solutions according to the cases (a) - (d)above. What we obtain is a (T, D) plane where each region is associated withthe type of equilibrium point and its stability properties. See Figure 8.6.


D

RETNEC

S A D D L E

S t a b l e

N o de or F o c u s

U n s t a b l e

N o d e or F o c u s

T

Figure 8.6: (T, D) plane. T 2 − 4D ≥ 0: Node. T 2 − 4D < 0: Focus.

Example 8.1.5 Consider the system

x1 = −3x1 + 2x2

x2 = −3x2.

The determinant of the coefficient matrix A =

[

−3 20 −3

]

is D = 9 > 0. The

trace is T = −6 < 0, and therefore T 2 − 4D = 0. Thus, we are in case (b) ofTheorem 8.5. That is, the origin is a stable node.

Indeed, the eigenvalues of A are λ1 = λ2 = −3, which indicates that the originis a stable node. See Figure 8.7

The real power of Theorem 8.5 is appreciated when the given two-dimensionalsystem depends on one or more parameters so that explicit expressions for theeigenvalues are too complicated to determine their sign. This often happenswhen the linear system results from a process of linearization applied to a givennonlinear system, as we will see in detail later on. As mentioned before, it maybe then more convenient to look at the determinant D and the trace T of thecorresponding coefficient matrix A, and decide accordingly about the propertiesof the origin. See also the equilibrium point analysis in Section 8.3.2.

Example 8.1.6 Consider the following system

x = (d− h)x+ (h− 1)yy = d2x+ (b− d)y,

(8.18)


−1 −0.5 0 0.5 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

x1

x 2

Figure 8.7: Phase portrait of Example 8.1.5

where b, d and h are certain nonzero parameters. In this case it is not possible tofind a simple explicit expression for the eigenvalues of the coefficient matrix interms of the given parameters. The trace and the determinant of the coefficientmatrix are given by

T = b− h, D = b(d− h) − dh(d − 1).

Without computing the eigenvalues explicitly we know that the origin will be asaddle when the determinant is negative, that is, whenever

b(d− h) < dh(d− 1).

Similarly, we can conclude that the origin will be a center when the determinantis positive and the trace is zero, that is, whenever b(d − h) > dh(d − 1) andb = h. Combining these two conditions, and using the fact that h 6= 0, theorigin will be a center whenever d− h > d(d− 1), or alternatively whenever

h < d(d− 2).

8.1.3 Stable, unstable, and center subspaces

We have already seen that the eigenvectors of the coefficient matrix in a dy-namical system x = Ax help determine the shape and direction of the solution


curves. In fact, something more fundamental is true: the eigenvectors associatedto eigenvalues with negative real part span together an invariant subspace withthe property that solutions starting on this subspace will stay on such subspacefor all time t. Even more, all solutions on that subspace will approach the ori-gin as t increases. Something similar happens with eigenvectors associated toeigenvalues with positive real part, but in this case solutions will go away fromthe origin. We want to formalize these ideas next.

Definition 8.6 A set S ⊂ Rn is called invariant with respect to a system x = Axif eAtS ⊂ S. In other words, for any initial vector x(t0) = x0 ∈ S, the solutionx(t) of x = Ax remains in S for all time t ≥ 0.

We can now formally define the stable, unstable and center subspaces of x = Ax.These subspaces are invariant under such system and are a central tool to studystability of a given dynamical system.

Definition 8.7 Let λj = aj + i bj , (j = 1, . . . , n) be the eigenvalues of An×nwith (generalized) eigenvectors uj = vj + iwj. Then we define

Es = span{ vj , wj : aj < 0 } stable subspace

Eu = span{ vj , wj : aj > 0 } unstable subspace

Ec = span{ vj , wj : aj = 0 } center subspace

(8.19)

The three subspaces are invariant with respect to the system x = Ax. Allsolutions in Es approach the equilibrium x = 0 as t → ∞; all solutions in Eu

approach the equilibrium x = 0 as t→ −∞.

Given an arbitrary system x = Ax, not all three subspaces in (8.19) are neces-sarily nonempty (to be more precise, they may contain only the origin), but inany case, it is always true that

Es ⊕ Ec ⊕ Eu = Rn. (8.20)

As usual, (see Definition 2.34), the notation ⊕ means that any element x ∈ Rn

can uniquely be written as x = u + v + w, with u ∈ Es, v ∈ Ec, w ∈ Eu.


Es

Eu

Figure 8.8: Invariant subspaces of Example 8.1.7

The whole space Rn is split into these three different subspaces, and the only

intersection between them is the origin.

Example 8.1.7 Let x = Ax, with A =

[

−1 −30 2

]

. Then the eigenvalues are

λ1 = −1, λ2 = 2, with corresponding eigenvectors v1 = [1 0]T , v2 = [−1 1]T .Thus, Es is the one-dimensional subspace spanned by v1; that is, Es coincideswith the x axis. Eu is the one-dimensional subspace spanned by v2; that is, Eu

is the line y = −x. There are no eigenvalues with zero real part, and thereforeEc is just the origin. See Figure 8.8.


−2 −1 01 −2 00 0 3

. Then the eigenvalues

are λ1,2 = −2 ± i, λ3 = 3, with eigenvectors u1,2 = v ± iw = [0 1 0]T ±i [1 0 0]T and u3 = [0 0 1]T . Thus, Es is the two-dimensional subspacespanned by v and w; that is, Es coincides with the xy plane. Eu is the one-dimensional subspace spanned by u3; that is, Eu coincides with the z axis. Ec

is just the origin. See Figure 8.9.


0 −1 01 0 00 0 2

. Then the eigenvalues are

λ1,2 = ±i, λ3 = 2, with corresponding eigenvectors u1,2 = v± iw = [0 1 0]T ±i [1 0 0]T , and u3 = [0 0 1]T . Thus, Ec is the two-dimensional subspace


Eu

Es

Figure 8.9: Invariant subspaces of Example 8.1.8

spanned by v and w; that is, Ec coincides with the xy plane. Eu is the one-dimensional subspace spanned by u3; that is, Eu coincides with the z axis. Es

is just the origin. See Figure 8.9 with Es replaced by Ec.

Remark 8.8 Solutions of x = Ax on the center subspace Ec do not necessarilystay bounded. See Exercise 8.20.

8.2 Nonlinear Dynamical Systems

We now consider systems of the form

x = f(x), (8.21)

where f : Rn −→ Rn is sufficiently smooth. The first and main difference be-tween nonlinear and linear systems, is that in general there is no closed formulafor a solution of (8.21) as observed in (8.3) for the linear case. In the greatmajority of cases, it will be too difficult or simply impossible to find an ex-act or analytical solution of (8.21), and we will need to make use of numericaltechniques to approximate a solution. However, without explicitly knowing thesolution, it is still possible to study in detail the qualitative behavior of suchsolution and decide on questions such as hyperbolicity, stability, asymptoticbehavior, invariant sets, etc.

8.2. NONLINEAR DYNAMICAL SYSTEMS 409

As mentioned before, the study of nonlinear systems can be divided in two parts:local and global. We start with a local approach through linearization, and thenstudy some aspects of global theory of nonlinear systems.

The main idea behind local theory is to study the nonlinear systems aroundcertain special sets such as equilibrium points and periodic orbits, by meansof linearization techniques, and then arrive to conclusions about the originalnonlinear system, but only in a certain vicinity of those special sets. Thus, thestrategy is to locate the equilibrium points and periodic orbits of the nonlinearsystem, apply linearization about these sets, study the resulting linear systems,and finally make conclusions about the qualitative behavior of the nonlinearsystems in some neighborhoods of those equilibria and periodic orbits.

One first classical result to introduce is the theorem on existence and uniquenessof solutions, as well as their dependence on the initial conditions. A clear proofof this statement can be found in [46]. The theorem says that if the function fis C1, then so is the solution u(t, y) in t and y, and for each fixed y, then u(t, y)is C2. We denote a neighborhood of a point x0, with radius δ as Nδ(x0).

Theorem 8.9 Existence, uniqueness, dependence on initial contidions.Let E ⊂ R

n be open, x0 ∈ E and f ∈ C1(E). Then,

(a) There exist a > 0, δ > 0 such that for all y ∈ Nδ(x0) the IVP

{

x = f(x)x(0) = x0

has a unique solution u(t, y) ∈ C1(G), where G = [−a, a] ×Nδ(x0).

(b) For each y ∈ Nδ(x0), u(t, y) ∈ C2( [−a, a] ).

The proof of the theorem uses the very important Gronwall’s inequality (Ex-ercise 8.27) and the fact that if f ∈ C1(E), then f is locally Lipschitz on E(Exercise 8.28). The central idea is to show that the successive approximations

u0(t, y) = y

uk+1(t, y) = y +∫ t

0 f(uk(s, y)) ds

converge uniformly to a function u(t, y) that satisfies

u(t, y) = y +

∫ t

0f(u(s, y)) ds,


so thatu(t, y) = f(u(t, y)) and u(t, y) = Df(u(t, y)) u(t, y).

Note: We say f is Lipschitz on E ⊂ Rn if

‖f(x) − f(y)‖ ≤ K‖x− y‖, (8.22)

for all x, y ∈ E and some constant K. And we say f is locally Lipschitz on E,if for each point x0 ∈ E there is some neighborhood Nǫ(x0) ⊂ E such that

‖f(x) − f(y)‖ ≤ K‖x− y‖,

for all x, y ∈ Nǫ(x0) and some constant K = K(x0).

As usual, if K < 1 in (8.22), we say f is a contraction.

Before we introduce two of the most important theorems in dynamical systems,we give the following definitions.

Definition 8.10 A point x0 ∈ Rn is called equilibrium point of (8.21) if

f(x0) = 0.

This definition implies that at such a point, x = 0, that is, the system remainsstatic (in equilibrium). In other words, if a solution curve reaches an equilibriumpoint at time t0, it will stay there for all time t > t0.

Definition 8.11 Let E ⊂ Rn be open, x0 ∈ E, f ∈ C1(E), and let φ(t, x0)

denote the solution of the IVP x = f(x), x(0) = x0 on some interval I. Then,for t ∈ I, the set of mappings φt : R

n → Rn defined by

φt(x0) = φ(t, x0)

is called the flow of the system x = f(x).

The function φt describes the motion of points x0 ∈ Rn along trajectories of

x = f(x).


Definition 8.12 The linearization of the system (8.21) is defined as

x = Ax, (8.23)

where A = Df(x), the Jacobian of f .

Definition 8.12 provides with a powerful tool to study nonlinear dynamical sys-tems locally. First we consider linearization around equilibria.

8.2.1 Linearization around an equilibrium point

For the technique of linearization introduced above, we will need to use a specialkind of equilibrium point x0. Then, the Jacobian will be evaluated at thisequilibrium point so that we obtain a linear system x = Ax, where A = Df(x0).

Definition 8.13 Let x0 be an equilibrium point of (8.21). Then, x0 is calledhyperbolic if the Jacobian A = Df(x0) has no eigenvalues with zero real part.

This definition implies that the eigenvalues associated with hyperbolic equilibriacan be located anywhere in the plane R

2 but the imaginary axis.


x1 = x1(6 − 2x1 − x2)x2 = x2(4 − x1 − x2).

(8.24)

The equilibrium points of (8.24) are (0, 0), (3, 0), (0, 4), (2, 2). The Jacobian of

f is given by A(x) = Df(x) =

[

6 − 4x1 −x1

−x2 4 − x1 − 2x2

]

. The four points are

hyperbolic; for instance, A(3, 0) =

[

−6 −30 1

]

, so that λ1 = −6, λ2 = 1.

Remark 8.14 Observe that in general, given an arbitrary matrix An×n, weexpect only a very small subset of its eigenvalues to “land” on the imaginaryaxis. That is, by restricting ourselves to study hyperbolic equilibria we are stillstudying the great majority of cases possible.


Recall that a homeomorphism is a function h : A → B that is a bijection, iscontinuous and whose inverse is also continuous.

We need to state two main theorems of dynamical systems which will allowus to perform linearization. We start by defining a differentiable manifold ofdimension n as a set that is locally homeomorphic to the usual Euclidean spaceRn. A differentiable manifold is in fact a topological space that generalizes the

intuitive and geometric notion of a curve or a surface. Consider for example theone-dimensional space R (say, the usual x axis). Then a differentiable manifoldhomeomorphic to it is the cubic parabola y = x3: it is a continuous deformationof the x axis.

Recall that for linear systems we have defined the linear vector subspaces Es, Ec

and Eu. As observed in Examples 8.1.7, 8.1.8 and 8.1.9, solutions starting onany of these subspaces at time t0 will stay there for all time t > t0 (they areinvariant). In particular, the solution will approach the equilibrium point ifstarting on Es and will move away from it if starting on Eu. The equivalentobjects for nonlinear dynamical systems are the so called stable, center andunstable manifolds W s, W c and W u respectively. They are not only invariantand have the same asymptotic behavior as the corresponding linear subspacesbut under certain conditions, they are also tangent to them. Without loss ofgenerality we consider the equilibrium point to be the origin.

Theorem 8.15 (Stable Manifold Theorem) Let E ⊂ Rn be open containing

the origin, f ∈ C1(E), and let φt be the flow of x = f(x). Suppose the originis a hyperbolic equilibrium point and that A = Df(0) has k eigenvalues withnegative real part and the remaining n − k eigenvalues have positive real part.Then,

(a) There exists a k-dimensional differentiable manifold S tangent to Es at theorigin, such that φt(S) ⊂ S, ∀t ≥ 0 and lim

t→∞φt(x) = 0, ∀x ∈ S.

(b) There exists an (n−k)-dimensional differentiable manifold U tangent to Eu

at the origin, such that φt(U) ⊂ U, ∀t ≤ 0 and limt→−∞

φt(x) = 0, ∀x ∈ U .

Sketch of proof. First write the system x = f(x) as

x = Ax+ F (x), (8.25)

where A = Df(0) and F (x) = f(x) − Ax, and consider the similarity transfor-


−1 −0.5 0 0.5 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Eu

Es

Wu

Ws

Figure 8.10: Schematic representation of the stable manifold theorem

mation

B = C−1AC =

[

P 00 Q

]

,

for some nonsingular C, where all the eigenvalues λ1, . . . , λk of the matrixPk×k have negative real part, and all the eigenvalues λk+1, . . . , λn of the matrixQ(n−k)×(n−k) have positive real part. Also observe that we can always choosesome α > 0 sufficiently small so that Re(λj) < −α < 0, for j = 1, . . . , k.

By letting y = C−1x, the system (8.25) can now be written as

y = By +G(y), (8.26)

where G(y) = C−1F (Cy), so that this new system is split into a stable and anunstable part. Next define the matrix functions

U(t) =

[

ePt 00 0

]

and V (t) =

[

0 00 eQt

]

,

and consider the integral equation

u(t, a) = U(t)a+

∫ t

0U(t− s)G(u(s, a))ds −

∫ ∞

t

V (t− s)G(u(s, a))ds. (8.27)

The key here is that if u(t, a) is a solution of this integral equation, then it isalso a solution of the differential equation (8.26). Thus, we seek to solve the


integral equation by successive approximations:

u(0)(t, a) = 0,

u(k+1)(t, a) = U(t)a+∫ t

0 U(t− s)G(u(k)(s, a))ds −∫ ∞tV (t− s)G(u(k)(s, a))ds.

(8.28)

The sequence {uk(t, a)} converges uniformly to a function u(t, a) that satisfiesthe integral equation, and therefore also the differential equation (8.26), and

|u(t, a)| ≤ K|a|e−αt, ∀ t ≥ 0, (8.29)

for some constant K. We are focusing our attention on the computation ofthe stable manifold. Observe from (8.27) that the last n − k components of ado not affect the computation and thus they are arbitrary. Therefore, we takea = [a1 · · · ak 0 · · · 0]T . For j = k + 1, . . . , n define the real functions

ψj(a1, . . . , ak) = uj(0, a). (8.30)

Then, the initial values yj = uj(0, a) satisfy

yj = ψj(y1, . . . , yk), j = k + 1, . . . , n. (8.31)

These are the equations that define a manifold S of (8.26), with the propertythat if y(0) = u(0, a) ∈ S, then y(t) = u(t, a) ∈ S for all t ≥ 0, and that y(t) → 0

as t→ ∞, according to (8.29). It can be shown that∂ψj

∂yi(0) = 0, for i = 1, . . . , k

and j = k+ 1, . . . , n, which implies that S is tangent to the stable subspace Es

of y = By at the origin. The stable manifold S is finally obtained from S bychanging back to x coordinates via x = Cy.

The existence of the unstable manifold is established similarly, by consideringthe system

y = −By −G(y).

The stable manifold of this new system will be the unstable manifold of thesystem (8.26). It is necessary to read the vector y as [yk+1 · · · yn y1 · · · yk]T inorder to determine the n− k manifold U . For full details on this proof, see [46].

�

Remark 8.16 The manifolds obtained in Theorem 8.15 are local. The globalmanifolds are obtained by extending the local ones in time. More precisely, theglobal stable and unstable manifolds respectively of x = f(x) are defined as

W s(0) =⋃

t≤0

φt(S) and W u(0) =⋃

t≥0

φt(U).



x1 = −x1

x2 = x21 − x2

x3 = x22 + x3.

Here, A = Df(0) =

−1 0 00 −1 00 0 1

, and F (x) = f(x) − Ax =

0x2

1

x22

. Since

the eigenvalues in A are already ordered, then B = A,C = I, and G(x) = F (x).We also have

U(t) =

e−t 0 00 e−t 00 0 0

, V (t) =

0 0 00 0 00 0 et

, a =

a1

a2

0

.

The successive approximations (8.28) are

u(0)(t, a) = [0 0 0]T ,

u(1)(t, a) = [e−ta1 e−ta2 0]T ,

u(2)(t, a) =

e−ta1

e−ta2

0

+

∫ t

0

e−(t−s) 0 0

0 e−(t−s) 00 0 0

0e−2sa2

1

e−2sa22

ds

−∫ ∞

t

0 0 00 0 0

0 0 e(t−s)

0e−2sa2

1

e−2sa22

ds

=

e−ta1

e−ta2

0

+

∫ t

0

0

e−(t+s)a21

0

ds −∫ ∞

t

00

et−3sa22

ds

=

e−ta1

e−ta2

0

+

0−e−2ta2

1 + e−ta21

0

−

00

−13e

−2ta22

=

e−ta1

(a21 + a2)e

−t − a21e

−2t

−13 e

−2ta22

.

Similarly,

u(3)(t, a) = [ e−ta1 (a21 + a2)e

−t − a21e

−2t − 15 a

41e

−4t + 12a

21(a

21 + a2)e

−3t

−13(a2

1 + a2)2e−2t ]T .


u(4)(t, a) = u(3)(t, a).

Since Df(0) has eigenvalues λ = −1,−1, 1, the stable manifold S = S is givenaccording to (8.31) by one equation in two variables x3 = ψ3(x1, x2), where (see(8.30))

ψ(a1, a2) = u3(0, a1, a2, 0) = −15 a

41 + 1

2a21(a

21 + a2) − 1

3 (a21 + a2)

2

= − 130a

41 − 1

6a21a2 − 1

3a22.

That is,

S = { [x1 x2 x3]T | x3 = − 1

30x4

1 −1

6x2

1x2 −1

3x2

2 }.

Remark 8.17 It should be clear that if the solution of the system x = f(x) isexplicitly available, then one can try to obtain the stable or unstable manifoldsdirectly, without having to do the successive approximations, by analyzing theasymptotic behavior of the solution. See Exercise 8.30.

The second main result is one of the most important theorems in the studyof nonlinear systems. It is the one that gives the conditions under which alinear system obtained by a linearization around an equilibrium point is locallyequivalent to the corresponding nonlinear system. The main condition imposedis hyperbolicity. The theorem states that we can study a nonlinear systemlocally, by considering the linearized system around hyperbolic equilibria.

Theorem 8.18 (Hartman-Grobman) Let x0 be a hyperbolic equilibrium of thenonlinear system (8.21). Then, in a neighborhood of x0, the system (8.21) andits corresponding linearization

x = Ax, (8.32)

where A = Df(x0), are equivalent; that is, there is a homeomorphism h thatmaps trajectories in (8.21) near x0 onto trajectories in (8.32).


x1 = x1

x2 = x21 − x2

(8.33)


−0.5 0 0.5−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4(a)

−0.5 0 0.5

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5(b)

Es

Eu

SU

Figure 8.11: Linearized and nonlinear systems of Example 8.2.3.

The origin is the only critical point. The Jacobian is Df(x) =

[

1 02x1 −1

]

and

the linearization of (8.33) is x = Ax, where A = Df(0, 0) =

[

1 00 −1

]

, with

eigenvalues λ1 = 1, λ2 = −1, so that (0, 0) is hyperbolic. This means we canapply Theorem 8.18 and study the nonlinear system (8.33) around (0, 0) throughits linearization. The eigenvector associated to the unstable eigenvalue λ1 isv1 = [1 0]T and the eigenvector associated to the stable eigenvalue λ2 = −1is v2 = [0 1]T . This means, the x-axis is the unstable direction, or Eu, andthe y-axis is the stable direction, or Es. It can be proved that in this case the(local) unstable manifold U is given by y = 1

3x2 and that the stable manifold S

coincides with Es. This gives a clear picture of how solutions of (8.33) behavearound the saddle point (0, 0). See Figure 8.11.

Observe in Example 8.2.3 that locally, the unstable manifold U is a continuousdeformation of the corresponding unstable subspace Eu, and that solutions ap-proach U (and move away from the origin) as t→ ∞, even if we start very closeto the stable manifold S. In this case, the only way to approach the origin is tostart exactly on S. The main point to observe is the local equivalence of both,the nonlinear and the linearized system around the equilibrium.


−4 −2 0 2 4 6 8 10−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

Figure 8.12: Solutions of (8.34)

Example 8.2.4 Let us study the local behavior of the system

x1 = x2

x2 = − sinx1 − cx2.(8.34)

There is an infinite number of equilibria: (0, 0), (±π, 0), (±2π, 0), etc. The Ja-

cobian is Df(x) =

[

0 1− cosx1 −c

]

and we have

Df(0, 0) =

[

0 1−1 −c

]

and Df(±kπ, 0) =

[

0 1−1 −c

]

or

[

0 11 −c

]

,

for k even or odd respectively. For (0, 0) or (±kπ, 0), with k even, the eigen-values are λ = (−c ±

√c2 − 4)/2. For c = 0, there will be circles around those

equilibria. For (±kπ, 0) with k odd, the eigenvalues are λ = (−c ±√c2 + 4)/2.

Thus, for arbitrary c, they are (hyperbolic) saddle equilibria. At these hyperbolicequilibria, we can apply Theorem 8.18 and study the nonlinear system (8.34)around (±kπ, 0) through its linearization: the unstable eigenvector (for c = 0)is v1 = [1 2]T and the stable eigenvector is v2 = [1 − 2]T . Observe that for(±kπ, 0) with k even, the Jacobian coincides with the Jacobian at (0, 0), andthere will also be circles around those equilibria. We have now a clear ideaabout the qualitative behavior solutions of (8.34) around the equilibrium points.See Figure 8.12.


8.2.2 Linearization around a periodic orbit

Just as we linearized a nonlinear system around equilibria, to understand at leastlocally the qualitative behavior of solutions of nonlinear systems, we can alsolinearize around periodic orbits. This will allow us to understand the behaviorof solutions around such periodic orbit. We start with the following

Definition 8.19 A solution of the system x = f(x) is called a periodic solutionof period τ if

x(t) = x(t+ τ), ∀ t ∈ R.

Note: Here, τ is in fact the minimum period, because nτ would also be a periodfor any n = 1, 2, . . . .

We will call periodic orbit to the set of points in the phase space that correspondsto a periodic solution.

We mentioned before that finding equilibria of x = f(x) can be accomplishedby solving the nonlinear system of algebraic equations f(x) = 0, say by usingNewton’s method. A periodic orbit of x = f(x) of period τ can be computed assolution of the boundary value problem

x = f(x), t ∈ [0, τ ]x(0) = x(τ).

(8.35)

In most cases, solutions to such boundary value problems must be approximatednumerically, say by multiple shooting.

Let A = Df(x) be the Jacobian of f , and consider the associated fundamentalmatrix solution Φ(t), as in Definition 7.15. Assume that the system x = f(x) hasa periodic solution of period τ . Then, the eigenvalues of the matrix M = Φ(τ)are called the Floquet multipliers of the periodic orbit. The matrixM is known asthe monodromy matrix. Just as the eigenvalues of the Jacobian at an equilibriumpoint determine its stability properties and the asymptotic behavior of solutionsaround it, the eigenvalues of the monodromy matrix determine the stabilityproperties of a periodic orbit and the asymptotic behavior solutions around it.Again, the central condition for linearization will be hyperbolicity.


Definition 8.20 A periodic orbit is called hyperbolic if it has exactly one Flo-quet multiplier of magnitude 1.

Remark 8.21 For any given periodic orbit, one of the Floquet multipliers isalways 1. Hyperbolicity guarantees that no other multiplier lies on the unit circle.On the other hand, multipliers are in general of the form µ = eλτ , where τ isthe period, and λ, in general complex, is called the Floquet exponent and is notunique: it is only determined modulo 2πi. However, they uniquely determine themagnitude of the Floquet multipliers, which ultimately determines the stabilityof the periodic orbits: multipliers with magnitude less than 1 are called stable,and those with magnitude greater than 1 are unstable.

Example 8.2.5 The planar system

x = x(1 − x2 − y2) − yy = y(1 − x2 − y2) + x

(8.36)

clearly has a solution of the form (x(t), y(t)) = (cos t, sin t), which is a periodicorbit of period τ = 2π. Details on Floquet multipliers and stability are given inExample 8.2.6.

The main question is whether we can apply a result like the Hartman Grobmantheorem for periodic orbits. The answer is yes, as long as the periodic orbit ishyperbolic. In other words, if the system x = f(x) is linearized around a hyper-bolic periodic orbit, then locally, the nonlinear system is qualitatively equivalentto the linearized one. This result allows us to study a nonlinear system around ahyperbolic periodic orbit by studying the corresponding linearized system. Evenmore, just as we have the stable manifold theorem for equilibria, we also haveone available for periodic orbits, with corresponding stable, center and unstablemanifolds W s,W c,W u respectively, and with the same properties as for equilib-ria: they are invariant and solutions starting on W s will approach the periodicorbit as t → ∞, solutions starting on W c will stay around the periodic orbit forall time t, and solutions starting on W u will move away from the periodic orbitas t→ ∞.

Theorem 8.22 Stable Manifold Theorem for Periodic Orbits Let E ⊂Rn be open containing a periodic orbit Γ, f ∈ C1(E), and let φt be the flow of


x = f(x). Suppose the periodic orbit is hyperbolic, and has k Floquet multiplierswith magnitude strictly less than one and n−k−1 with magnitude strictly greaterthan one. Then,

(a) There exists a (k+1)-dimensional differentiable manifold S such that φt(S) ⊂S, ∀t ≥ 0 and lim

t→∞φt(x) = Γ, ∀x ∈ S.

(b) There exists an (n − k)-dimensional differentiable manifold U such thatφt(U) ⊂ U, ∀t ≤ 0 and lim

t→−∞φt(x) = Γ, ∀x ∈ U .

Remark 8.23 If we denote the Floquet exponents with λj = aj + ibj , then onecan define the stable, center and unstable subspaces of the corresponding periodicorbit as in (8.19), using the associated generalized eigenvector of the monodromymatrix. These subspaces are tangent to the corresponding manifolds defined inTheorem 8.22.


x = x(1 − x2 − y2) − yy = y(1 − x2 − y2) + xz = 3z

(8.37)

This system has a periodic orbit γ(t) := (cos t, sin t, 0), and we want to linearizethe system around γ. First, we need to compute the Jacobian:

Df(x, y, z) =

1 − 3x2 − y2 −1 − 2xy 01 − 2xy 1 − x2 − 3y2 0

0 0 3

. (8.38)

Evaluating this at γ, we get

A(t) = Df(γ(t)) =

−2 cos2 t −1 − sin 2t 01 − sin 2t −2 sin2 t 0

0 0 3

.

Then, the linearization of (8.37) is

x = A(t)x, (8.39)

with fundamental matrix solution

Φ(t) =

e−2t cos t − sin t 0e−2t sin t cos t 0

0 0 e3t

.


−2

−1

0

1

2 −2

−1

0

1

2

−0.05

0

0.05

S

U

Figure 8.13: Solutions and invariant subspaces of (8.37)

The Floquet multipliers of γ are computed as the eigenvalues of the monodromymatrix Φ(2π), which are µ1 = 1, µ2 = e−2 ≈ 0.1353 and µ3 = e3 ≈ 20.0855.Then, γ is hyperbolic, because it has exactly one multiplier of magnitude one (italso has one stable multiplier, µ2, and one unstable multiplier, µ3). From The-orem 8.22, there is a two-dimensional stable manifold S, in this case coincidingwith the xy plane (excluding the origin), and a two-dimensional unstable man-ifold W u, which is a unit cylinder: solutions spiral on the walls of the cylinderand away from the periodic orbit. See Figure 8.13.

We have introduced some ideas on local analysis of nonlinear systems aroundhyperbolic equilibria and hyperbolic periodic orbits. When hyperbolicity is vi-olated, study of nonlinear systems becomes more complex, and more advancedmathematical tools are needed, such as center manifolds, which is beyond thescope of this book. The reader is referred to books on dynamical systems suchas [12], [14] and [23]. Here we want to introduce two special sets of solutionsthat illustrate some global behavior of nonlinear systems: connecting orbits andchaotic solutions.


8.2.3 Connecting orbits

In Example 8.2.4 we have already seen some global solutions of a nonlinearsystem. Those solutions connecting (−π, 0) to (π, 0), those connecting (π, 0) to(3π, 0), etc, are very special solutions commonly known as heteroclinic orbits.In the particular case where a solution connects back to the same equilibriumpoint, it is called homoclinic orbit. Homoclinic and heteroclinic are also knownas connecting orbits. They not only connect equilibrium points, but they canalso connect an equilibrium to a periodic orbit and two periodic orbits as well.This type of solutions are very important in the study of dynamical systems asthey form what is known as separatrices: solutions that serve as border curvesbetween several solutions of the system. Observe in Figure 8.12 how the het-eroclinic connections between (−π, 0), (π, 0), (3π, 0), etc. have circle solutionsinside them, and other types of curves outside them, so that when certain sepa-ratrices are found, we can have a good understanding of the qualitative behaviorof solutions, and we are not necessarily restricted to only local setting anymore.

There is an increasing interest on connecting orbits mainly due to several im-portant applications. For example, in [10] connecting orbits are identified in awater-wave model, in [33], [34] the authors find homoclinic and heteroclinic con-nections in a model of celestial mechanics and they show how this may lead tospace exploration with prescribed itineraries, and in [47] traveling waves fromseveral applications are computed using connecting orbits. We want to givesome basic ideas of this special type of solutions. For a detailed treatment ofconnecting orbits and applications, see e.g. [5], [16] and [17].

We need to start by considering a more general form of a nonlinear system, onethat includes the presence of one or more real parameters in the vector field.Namely, we consider the parameter-dependent nonlinear systems

x = f(x, λ), x(t) ∈ Rn, λ ∈ Rp, (8.40)

where f : Rn × Rp −→ Rn is assumed to be sufficiently smooth. Let M−(λ)be either a hyperbolic equilibrium y−(λ) or a hyperbolic periodic orbit γ−(λ) of(8.40), and let M+(λ) be a hyperbolic periodic orbit γ+(λ) of (8.40). We will usethe notation y+(t, λ) to denote the periodic solution of period τ+ correspondingto γ+(λ), and similarly y−(t, λ) will be the periodic solution of period τ− relativeto γ−(λ), if M−(λ) = γ−(λ). As noted before, equilibrium points can be foundby solving the system f(x, λ) = 0, and we will assume them to be known.However, the periodic orbits will need to be computed as part of the problem.


A solution x(t, λ), t ∈ R of (8.40) is called a connecting orbit from M−(λ) toM+(λ) if

dist (x(t, λ), M±(λ)) −→ 0 as t −→ ±∞. (8.41)

Firstly, notice that now we are dealing with asymptotic solutions, those that aredefined as t increases to infinity. Secondly, since the system (8.40) is autonomous(the vector field f does not explicitly depend on t), if x(t, λ) is a solution, thenalso x(t+σ, λ) is a solution, for any σ ∈ R. Then, an extra condition is necessaryfor the solution to be uniquely determined. Then, we impose a so called phasecondition

ψ(x, y−, y+, λ) = 0.

The central idea in all methods to compute connecting orbits is to truncate thereal line to a finite and sufficiently large interval [T−, T+], T− < 0 < T+, and im-pose boundary conditions at T±. A key observation is that the connecting orbitmust leave M−(λ) along its unstable manifold and enter M+(λ) along its stablemanifold. We know that these manifolds are tangent to the unstable subspaceEu−(λ) of M−(λ) and to the stable subspace Es+(λ) of M+(λ), respectively.

By using the so-called projection boundary conditions [5], [16], the problem ofcomputing the connecting orbit is transformed into the problem of solving thefollowing boundary value problem

x = f(x, λ), T− ≤ t ≤ T+

L−(λ)(x(T−) − y−(0)) = 0 , L+(λ)(x(T+) − y+(0)) = 0 ,ψ(x, y−, y+, λ) = 0 ,

(8.42)

where L− and L+ are smooth functions of λ, and span Ecs− (λ) and Ecu+ (λ), thecenter-stable and center-unstable subspaces of M−(λ) and M+(λ) respectively.The vector x(T−) − y−(0) lies on the unstable subspace Eu of M−(λ), andsimilarly x(T+) − y+(0) lies on the stable subspace Es of M+(λ). Thus, whenthey are multiplied by L−(λ) and L+(λ) respectively, their products in (8.42)vanish, by orthogonality. Also, in 8.42, ψ corresponds to the truncated versionof the phase condition.

To find the matrix functions L− and L+ in (8.42), we make use of a veryimportant matrix factorization introduced in Chapter 4: Schur factorization. Inthe case of an equilibrium, we perform the Schur factorization of the Jacobianevaluated at that point, and in the case of a periodic orbit, we compute the Schurfactorization of the monodromy matrix. The starting point is the ordered blockSchur factorization (4.39), but this time our matrix A depends on a parameter


λ, and we want to perform a smooth ordered block Schur factorization of A(λ),with factors as smooth as the matrix A(λ). See [15].

Thus, assume A(λ) is the n × n matrix function representing the Jacobian (orthe monodromy matrix), at a hyperbolic equilibrium (or periodic orbit), and weare interested in finding L−(λ). Assume further that A(λ) has ns eigenvalueswith negative real part (or ns+1 eigenvalues with magnitude less than or equalto one). Then, the (ordered block) Schur factorization gives

QT (λ)A(λ)Q(λ) = R(λ) =

[

R11(λ) R12(λ)0 R22(λ)

]

, (8.43)

where R11(λ) is square of order ns (or ns+1), and if µ is an eigenvalue of R11(λ),then Re(µ) < 0 (or |µ| ≤ 1). Then, we partition the orthogonal matrix Q(λ) as

Q(λ) = [Q1(λ) Q2(λ) ],

where Q1(λ) is of order n× ns (or n× (ns + 1) ), and whose columns span thestable (or center-stable) subspace associated to A(λ). Thus,

L−(λ) := Q1(λ).

We follow a similar idea to compute L+(λ). The main point is that the orderedSchur factorization allows us to locate the eigenvalues we are interested in atthe upper left block of R, and from there we can identify Q1(λ) and thereforeL−(λ) or L+(λ).

Now let W cu− (λ), respectively W cs

+ (λ), be the center-unstable manifold ofM−(λ),respectively the center-stable manifold of M+(λ). Suppose that there is a con-necting orbit γ, connecting M− and M+, then, we must have γ ⊂ W cu

− ∩W cs+ .

We expect the connecting orbit γ to be isolated if for the tangent spaces at z(t)we have

Tz(t)Wcu− ∩ Tz(t)W

cs+ = Tz(t)γ = span {z(t)}, ∀ t ∈ R, (8.44)

and to persist if the intersection is transversal, i.e.

Tz(t)Wcu− + Tz(t)W

cs+ = Rm+p , ∀ t ∈ R, (8.45)

where z = (x, λ).

For each given λ, let us write nu−, nc−, and ns−, for the dimensions of the unstable,center, and stable manifolds of M−(λ), and analogously we will write nu+, nc+,


and ns+ relatively to M+(λ). We will have that nc± = 0 for an equilibrium point,and nc± = 1 for a (hyperbolic) periodic orbit. With this notation, it is possible toestablish a fundamental relation between number of parameters and dimensionsof W cs,cu

± from (8.44) and (8.45), which reads [5]:

p = nu+ − nu− − nc− + 1. (8.46)

The main result in the theoretical setup is that the connecting orbit problem

x = f(x, λ) , −∞ < t <∞ ,lim

t→−∞dist(x(t, λ),M−(λ)) = 0 , lim

t→+∞dist(x(t, λ),M+(λ)) = 0 ,

ψ(x, y−, y+, λ) = 0 ,

(8.47)

is well posed if and only if the manifolds W cu− and W cs

+ intersect transversallyin the sense of (8.44)-(8.45).

Connecting orbits can be efficiently computed by numerically solving the bound-ary value problem (8.42). Under appropriate conditions, the error, which re-sults from truncating the interval of integration (−∞,∞) to a finite intervalJ := [T−, T+], decays exponentially. To state this result formally, we first givethe following

Lemma 8.24 Let F : Bδ(w0) → Z be a C1 mapping from some ball of radiusδ in a Banach space W into some Banach space Z. Assume that F ′(w0) is anhomeomorphism and that for some constants c1, c2 we have

‖F ′(w) − F ′(w0) ‖ ≤ c2 < c1 ≤ ‖F ′(w0)−1‖−1, ∀w ∈ Bδ(w0) , (8.48)

‖F (w0) ‖ ≤ (c1 − c2) δ. (8.49)

Then F has a unique zero wc in Bδ(w0) and

‖w0 − wc ‖ ≤ (c1 − c2)−1‖F (w0) ‖ , (8.50)

‖w1 −w2 ‖ ≤ (c1 − c2)−1‖F (w1) − F (w2) ‖ , ∀ w1, w2 ∈ Bδ(w0). (8.51)

Next, we need to define the following spaces

W := C1(J,Rn) ×Rp, Z := C(J,Rn) ×Rnc−

+ns− ×Rn

u++1.

For α, β > 0, their norms are defined as


‖(x, λ)‖W = supt∈J

−

‖x(t)‖eαt + supt∈J+

‖x(t)‖e−βt + ‖λ‖,

‖(y, r−, r+)‖Z = ‖(y, r−)‖Z1+ ‖(y, r+)‖Z2

, where

(8.52)

‖(y, r−)‖Z1= sup

t∈J−

‖y(t)‖eαt + ‖r−‖, and

‖(y, r+)‖Z2= sup

t∈J+

‖y(t)‖e−βt + ‖r+‖, ‖ · ‖ = ‖ · ‖∞,(8.53)

where J− = [T−, 0] and J+ = [0, T+]. With these norms, W and Z becomeBanach spaces (that is, complete normed vector spaces: every Cauchy sequenceis convergent). Anticipating the asymptotic convergence of x(t) to y(t) withrate ǫ > 0 we impose the condition that for some constant C, ‖x(t)−y±(t)‖ ≤Ce−ǫ|t|, as t→ ±∞.

The following theorem says that the error in approximating the connecting orbitis bounded by the projection boundary conditions.

Theorem 8.25 Let (8.44), (8.45) hold, and let (x, λ) be an orbit connectingeither a hyperbolic equilibrium point y−(λ) or a hyperbolic periodic orbit γ−(λ),to a hyperbolic periodic orbit γ+(λ). Consider (8.42) and assume that f ∈C2(Rn+p, R

n), and that L± are C1 (in λ). Then, there exists δ > 0 sufficientlysmall and C > 0, such that, for sufficiently large interval of integration J =[T−, T+ ], the boundary-value problem (8.42) has a unique solution (xJ , λJ) ina ball of radius δ in W . Moreover, the following estimate holds

‖(xJ , λ) − (x|J , λ)‖W ≤ C(

‖L−(λ)(x(T−) − y−(s(T−))) ‖+ ‖L+(λ)(x(T+) − y+(s(T+))) ‖

)

.(8.54)

The above theorem can be proved by applying Lemma 8.24 to w0 = (x|J , λ) and

F (x, λ) = (x−f(x, λ), L−(λ)(x(T−)−y−(s(T−))), L+(λ)(x(T+)−y+(s(T+))) ).

Finally, we state the main theorem on connecting orbits, which establishes ex-istence and uniqueness of solutions as well as exponentially decaying errors inthe approximation.


Theorem 8.26 Under the assumptions of Theorem 8.25, for J = [T−, T+]sufficiently large, we have

‖(xJ , λ) − (x|J , λ)‖W ≤ C e−2min (µ−|T

−| , µ+T+ ) . (8.55)

In (8.55), 0 < µ− < Re µ, for all unstable eigenvalues µ of the Jacobianfx(y−(λ)) (if M−(λ) = y−(λ)), or all unstable Floquet exponents of the mon-odromy relative to γ−(λ) (if M−(λ) = γ−(λ)). Also, 0 < µ+ < -Re µ, for allstable Floquet exponents µ associated to the periodic orbit γ+(λ).

Proof. The key tools to use are corollaries from the stable manifold theorems forequilibria and periodic orbits (see e.g. [27]), which state that solutions starting inthe corresponding unstable or stable manifold, sufficiently near the equilibriumor the periodic orbit, approach them exponentially fast, as t → −∞ or t → ∞respectively. Moreover, in the case of a periodic orbit, the motion along theconnecting orbit is synchronized with that on the periodic orbit (convergence inasymptotic phase). For the boundary condition at an equilibrium, there existsT1 < 0, such that [4]

L−(λ)(x(t) − y−(s(t))) = O( e−2µ−t ) , for t ≤ T1 . (8.56)

Next we give the proof for the exponential decay of the error for the boundarycondition at the periodic orbit γ+. If 0 < µ+ < −Re µ, for all characteristicexponents µ with negative real part of the periodic orbit γ+(λ), then there existsT2 > 0 such that for all t ≥ T2 ,

x(t) − γ+(λ) = O(e−µ+t) . (8.57)

Then, for any t ≥ T2 there is always a time shift s(t) : 0 ≤ s(t) ≤ τ+, such thatx(t) − y+(s(t)) = O(e−µ+t). With this, by a Taylor expansion, we get

L+(λ)(x(t) − y+(s(t))) = L+(λ)(y+(s(t)) − y+(s(t)))

+ L+(λ)(y+(s(t)) − y+(s(t))) (x(t) − y+(s(t))) + O( ‖x(t) − y(s(t))‖2 ‖).

Therefore,L+(λ)(x(t) − y+(s(t))) = O( ‖x(t) − y+(s(t))‖2 ) (8.58)

and by (8.57),L+(λ)(x(t) − y+(s(t))) = O( e−2µ+t ). (8.59)

As for the periodic-to-periodic case, if 0 < µ− < −Re µ for all unstable Floquetexponents of the periodic orbit γ−(λ), then there exists a T1 < 0 such that forall t ≤ T1,

x(t) − γ−(λ) = O(e−µ+t), and (8.60)


proceeding in a similar way as we did for the periodic orbit γ+, we can obtain(for a time shift s(t)))

L−(λ)(x(t) − y−(s(t))) = O( e−2µ−t ). (8.61)

Combining (8.56), or (8.61), and (8.59) with inequality (8.54) from Theorem8.25, we get the sought result.

�

With the theoretical background well established, algorithms for the numericalcomputation of connecting orbits can be constructed. The main part of such acode deals with solving the nonlinear system x = f(x, λ) as a boundary valueproblem:

x = (T+ − T−)f(x, λ) , 0 ≤ t ≤ 1 ,

L−(λ)(

x(0, λ) − y−(λ))

= 0 ,

L+(λ)(

x(1, λ) − y+(0, λ))

= 0 ,

(8.62)

and calling a subroutine to compute the periodic orbit, for the current givenvalue of λ = λ:

y+ = τ+f(y+, λ) , 0 ≤ t ≤ 1 ,

y+(0) = y+(1) ,

σ(y+, λ) = 0 ,

(8.63)

where σ = 0 is a phase condition for the periodic orbit, serving the role of ψ = 0in (8.42), and the intervals on both systems are rescaled to [0, 1]. See [16] fordetails.

Example 8.2.7 We consider the well-known Lorenz equations. This is a systemthat has been extensively studied because of the wide range of solution behaviors itshows, including chaotic solutions as well as homoclinic and heteroclinic orbits.Originally, the system was proposed by Lorenz as a simple model for weatherprediction. The system is

x1 = σ (x2 − x1)x2 = λx1 − x2 − x1x3

x3 = x1x2 − b x3,(8.64)

where we take σ = 10, b = 83 and treat λ as a free parameter. There exists

a connection from the origin to a periodic orbit as the result of the transversal


−100−50050100−200−1000100200

0

50

100

150

200

250

300 b = 5.276666

b = 7.036666

b = 8.976666

Figure 8.14: Connections in (8.64). λ = 41.91, 70.12, 151.92

intersection of the one-dimensional unstable manifold of the origin with the two-dimensional center-stable manifold of the periodic orbit for λ = 24.057900. Thecomputed periodic orbit has period τ = 0.677171 and Floquet multipliers µ1 = 1,µ2 = 1.029332, µ3 = 0.000092. Thus, it is an unstable periodic orbit, andtherefore finding a connection to it is not an easy task.

It is possible to perform what is known as continuation of solutions, that is, oncea solution is located, allow the parameters to vary and compute other solutionsfor those values of the parameters. In this case, we do continuation on theb parameter (and λ changes correspondingly). In Figure 8.14 we show a fewconnecting orbits obtained by smooth continuation, and in Table 8.2.7 we showhow the periods and the Floquet multipliers change as the parameters vary.

For illustration, we show the Jacobian matrix J at the origin (0, 0, 0) and itsblock Schur factorization involved in the computation of the connecting orbit forλ = 2.7566666666.

QTJ Q = QT

−10 10 0λ −1 00 0 −8

3

Q


b λ Period Floquet multipliers

2.7566666666 24.4872943345 0.6633333236 0.00010553211.0317331504

5.2766666666 41.9064116742 0.4301691358 0.00038757101.0945902025

8.9766662563 151.9179506324 0.2206728489 0.00811056221.5012230948

11.9766664585 861.7281515695 0.1304312721 0.00521482829.5770852838

13.4416464434 4829.6606254594 0.1000748120 0.000004968417715.58156289

Table 8.1: Periods and multipliers for Lorenz system (8.64)

=

−12.41495963 0.00000000 −7.243333330.00000000 −2.66666666 0.00000000

0.00000000 0.00000000 1.41495963

,

where Q = [Q1 Q2 ] =

−0.97205634 0.00000000 0.234747680.23474768 0.00000000 0.972056340.00000000 1.00000000 0.00000000

.

Then, the matrix L(λ) := Q1 spans the stable subspace of the equilibrium point(0, 0,0) at λ = 2.7566666666.

Example 8.2.8 To compute some periodic-to-periodic connections, we considerthe following system

x = (1 − w)y + wx(1 − x2)

y = (1 − w)(

−x+ λ(1 − x2)y)

+ w(z − γ)

z = (1 − w)z(

(z2 − (1 + γ)2)

+ w[

−y + γ + λ(

1 − (y − γ)2)

(z − γ)]

w = w(1 −w),

(8.65)where γ = 3 + λ, and λ is a real positive parameter. This system is a homotopyfrom w = 0 to w = 1 which in essence takes us to two planar systems living inthe (x, y) and (y, z) planes, respectively.


−3−2

−10

12

3

−4

−2

0

2

4

60

1

2

3

4

5

6

Figure 8.15: (8.65): Periodic-to-Periodic, λ = 0.5

In the (x, y) and (y, z) planes the equations reduce to those of van der Pol os-cillators. As it is well known, these oscillators have attracting periodic orbits(restricted to their respective planes). There are several heteroclinic connectionsbetween the two limit cycles of these van der Pol oscillators, and here we are in-terested in computing (and continuing) the one from z = 0, w = 0, call it γ−, tothat with x = 0, w = 1, call it γ+. A simple computation shows that associatedto both γ± there are two multipliers less than 1, one equal to 1, and one greaterthan 1. Therefore, we have nu± = 1, nc± = 1, and ns±=2. The balance (8.46) willgive us p = 0, hence there are no free parameters in the problem. In Figure 8.15we show the connection for λ = 1/2, and in Figure 8.16 we show the secondcomponents of several of these connecting orbits obtained from continuation.

8.2.4 Chaos

In the area of differential equations and dynamical systems, chaos theory isa relatively new area of study. It has grown in importance not only becauseof the challenging mathematics behind it, but also because of its wide rangeof important applications in areas such as physics, physiology, fractals, secure


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−4

−3

−2

−1

0

1

2

3

4

5

6

λ=0.5 λ=0.75 λ=1

Figure 8.16: (8.65): Second component of connecting orbits

communication and fluid dynamics. Recall that given the linear system

x = Ax, x(0) = x0, (8.66)

its solution is explicitly given by

x(t) = eAtx0. (8.67)

This means that the solution is known for any time t ≥ 0 and everything ispredictable, so that no chaotic behavior is possible. However, when we considernonlinear systems

x = f(x), x(0) = x0, (8.68)

there is no closed formula like the one in (8.67) for its solution. Thus, even ifthe vector field f is very smooth, the solution x(t) may be unbounded or evenunpredictable in general; some strange solutions may appear which apparentlycannot be entirely and precisely described; this is one of the main differencesbetween linear or nonlinear systems. Chaos theory tries to find and describe theunderlying order in apparently random data. Chaotic behavior is also relatedto sensitive dependence on initial conditions: the smallest change in the initialconditions can drastically change the long-term behavior of a system. This isexactly what Edward Lorenz experienced in the 1960’s for the first time when


−20 −15 −10 −5 0 5 10 15 20

−50

0

500

5

10

15

20

25

30

35

40

45

50

Figure 8.17: Strange attractor of (8.69)

he was studying a dynamical system for weather prediction. He realized thatsmall errors in the input data (a finite number of digits of precision) caused thesystem to behave entirely differently; this made him conclude that it is simplyimpossible to predict the weather accurately. For a detailed study of the Lorenzsystem see [51].

For a better illustration on the ideas of chaos, we briefly present some simplesystems where a variety of solutions, including chaotic behavior is observed.

Lorenz system. We study the system considered in Example 8.2.7

x1 = σ (x2 − x1)x2 = λx1 − x2 − x1x3

x3 = x1x2 − β x3,(8.69)

with the parameter values σ = 10, λ = 28, and β = 8/3. The equations rep-resent a simplified model of weather as fluid motion in the atmosphere drivenby thermal buoyancy, known as convection. The variable x1 measures the rateof convection overturning, the variable x2 measure the horizontal temperaturevariation, and the variable x3 measures the vertical temperature variation. Theparameter σ represents the ratio of the viscous to thermal diffusivities, λ repre-sents the temperature difference and is the control parameter. There is a greatvariety of solutions to the system depending on the values of λ (see for instanceExample 8.2.7). For λ = 28, solutions are pulled to what is known as a strange


0 20 40 60−20

−15

−10

−5

0

5

10

15

20FIRST COMPONENT

0 20 40 600

5

10

15

20

25

30

35

40

45

50THIRD COMPONENT

Figure 8.18: Solution components vs. time of (8.69)

attractor. See Figures 8.17 and 8.18. Even more, the phase portrait will lookvery different if the initial condition is changed slightly. See Exercise 8.52.

The classical solutions that we know are the steady states or equilibrium points,in which the values of the coordinates never change after reaching those points,and the periodic solutions, where no matter how simple or complicated thetrajectory is, the system goes into a loop repeating itself indefinitely. But inthis new kind of solution, the trajectory does not settle down to a steady stateand it is not periodic either. There is no way to predict exactly the path of thesolution as time increases, but we know that it will stay around this strange set.

Chua’s circuit. This is the simplest electronic device and model that exhibitscomplex behavior and a variety of chaotic phenomena. This is what makes itvery popular and it is also considered to be the paradigm for chaos. As Figure(8.19) shows, the circuit introduced by Leon Chua in the early 80’s, consists oftwo capacitors C1, C2, one resistor R, one inductor L, and one non-linear resistor(the Chua’s diode). Observe that the main difference between the Chua’s circuitand the RLC circuit (see Section 7.1.9) is the presence of the non-linear resistor.

If we let x1 = V1, x2 = V2 and x3 = IL, the system defining the Chua’s circuit is

x1 = α(x2 − h(x1))x2 = x1 − x2 + x3

x3 = −βx2,(8.70)

where h(x) = 27x− 1

28 (|x + 1| − |x− 1|). This function h is not smooth, but it


+

−

+

−

+

−

R

L

L

V VC

V

IIR

R

1 1C

2 2−

+

−

+

Figure 8.19: Chua’s circuit

can be approximated with a smooth function, by taking |x| ≈ 2π

arctan(10x).

The chaotic behavior of this system has been observed not only through math-ematical analysis and computer simulation but also in laboratory experiments.As remarked before, it is a very useful system because it is quite simple and yetit exhibits several special solutions: homoclinic to the origin, torus breakdown,Hopf bifurcations, period doubling, stochastic resonance, strange attractors, etc.In particular, a double-scroll attractor is shown in Figure 8.21. .

We remark that one interesting potential application of chaos is in secure com-munication. The idea is to encode a message within a (high dimensional) chaoticdynamics through small perturbations of a control parameter. Chaos is used toencrypt messages so that the transmitted signal is the sum of a chaotic signal anda given message, which can be reconstructed by the receiver, when synchronizedwith the sender.

8.2.5 Bifurcations

In most applications, the vector field f depends on one or more parameters,that is, the dynamical systems take the form

x = f(x, µ), (8.71)

with µ ∈ Rp, for some positive integer p. In general, solutions of (8.71) will

vary as the parameters vary, but most importantly, there are special values ofthe parameters, say µ = µ0, such that an arbitrarily small variation around µ0


−4−2

0

−0.50

0.5−5

0

5

(a)

−4−2

0

−0.50

0.5−5

0

5

(b)

02

4

−0.5

0

0.5−5

0

5

(c)

02

4

−0.5

0

0.5−5

0

5

(d)

Figure 8.20: (a),(b) Period-doubling and (c),(d) Strange attractor of (8.70)

−3−2

−10

12

3

−0.5

0

0.5

−5

0

5

Figure 8.21: Chua’s double-scroll attractor


will cause drastic changes in the qualitative behavior of the solutions. This phe-nomenon known as bifurcation has grown in importance due mainly to its abun-dant applications in different fields, such as macroeconomic systems, physics,space exploration, biology, physiology, etc. The purpose of this section is topresent the basic ideas around the changes in qualitative behavior (in particu-lar, stability) of solutions as a parameter varies.

The concept of bifurcation is closely related to that of stability and to hyper-bolicity. We want to start by introducing the concept of structural stability. Tothis end, for a vector field f ∈ C1(E), where E is an open subset of R

n, weintroduce the C1 norm

‖f‖1 = supx∈E

‖f(x)‖ + supx∈E

‖Df(x)‖, (8.72)

where in the right hand side, the first is the Euclidean norm, and the second isa matrix norm.

Definition 8.27 The dynamical system (8.71) is called structurally stable ifthere exists an ǫ > 0 such that for all g ∈ C1(E), with

‖f − g‖1 < ǫ,

f and g are equivalent in the sense that there is an homeomorphism that mapstrajectories of (8.71) onto trajectories of x = g(x, µ).

In simple words, this means that (8.71) is structurally stable if the qualitativebehavior does not change for all nearby vector fields, i.e. small variations in fdo not imply changes in such properties like stability or dimensions of stable orunstable manifolds. The homeomorphism property means that we can go froma trajectory of (8.71) to one of x = g(x, µ) or back, by a continuous deformation

One problem that has attracted much attention is on how to characterize or iden-tify structurally stable systems. This is a very important question to solve, assuch stability property is a fundamental one to consider when studying systemsmodeling real world problems. For instance, when considering macroeconomicmodels, robustness of inferences about the behavior of the system becomes criti-cally dependent on the sensitivity of the system to small changes in parameters.

For the case of two-dimensional systems the famous Peixoto’s Theorem givesnecessary and sufficient conditions for a dynamical system to be structurallystable in terms of hyperbolicity, wandering sets and connecting orbits.


Unfortunately, there is no counterpart to Peixoto’s theorem in higher dimen-sions. However, it is possible to give sufficient conditions for a system to bestructurally stable for any dimensions. To this end, we need two definitions;first we define a nonwandering point as the one that stays inside an arbitraryneighborhood of it under the flow defined by (8.71). The nonwandering set of(8.71) is the set of all nonwandering points of (8.71). Common examples ofnonwandering sets are equilibria and periodic orbits.

Definition 8.28 Two differentiable manifolds M and N in Rn are said to in-

tersect transversally if for every point p ∈M ∩N , the tangent spaces satisfy

TpM ⊕ TpN = Rn.

There is a set of sufficient conditions that a system x = f(x, µ) has to satisfyfor it to be structurally stable. Such a system is known as Morse-Smale system.These conditions are:

1. the number of equilibrium points and periodic orbits is finite and each ishyperbolic,

2. all stable and unstable manifolds which intersect do so transversally,

3. the nonwandering set consists of equilibrium points and periodic orbitsonly.

Theorem 8.29 (Palis and Smale) Every Morse-Smale system is structurallystable.

The conditions for a Morse-Smale system are very similar to those in Peixotos’stheorem, but more general, and Theorem 8.29 applies to dimensions higher thantwo. Unfortunately only, the converse of the theorem is not necessarily true fordimensions higher than three.

When systems are not structurally stable, say, the equilibrium points or periodicorbits are not hyperbolic, then we can expect bifurcation phenomena to happen,and in fact here we mainly consider nonhyperbolicity as source of bifurcations.But again, this does not mean that systems with only hyperbolic equilibria andperiodic orbits are structurally stable; all conditions above must me satisfied.


Now we need to give a definition of bifurcation. Consider again the system(8.71). We can define bifurcation as the phenomenon representing the sud-den appearance of qualitatively different solutions as the parameter(s) is (are)slightly varied. For instance, an equilibrium point at certain value of the pa-rameter λ suddenly gives way to a set of periodic orbits, after an arbitrarilysmall change of the parameter. More precisely, a value λ = λ0 for which thesystem (8.71) is not structurally stable is called a bifurcation value, and thecorresponding (x0, λ0) is called bifurcation point.

A great amount of work has been performed in devising algorithms to numeri-cally locate bifurcations of nonlinear systems, and to identify the type of bifur-cation. Probably the most reliable software available for the numerical study ofbifurcations is AUTO [18], originally developed by E. Doedel. Also worth men-tioning is XPPAUT [20], which is based in AUTO, but a little more user-friendly,and more recently Matcont [22].

Next, we present some of the most basic bifurcations of nonlinear systems.Here we illustrate such bifurcations by using the so-called bifurcation diagrams,where typically the horizontal axis is one of the parameters and the vertical axisis the norm of the solution. Thick solid curves represent stable equilibria anddash lines represent unstable equilibria. Stable periodic orbits are representedby small solid circles whereas unstable periodic orbits are represented by emptycircles. The bifurcation diagrams shown here were obtained using XPPAUT.

Transcritical bifurcations.

This type of bifurcation is characterized by an exchange of stability at bifurca-tion values (a stable equilibrium becomes unstable and an unstable one becomesstable), and by the presence of nonhyperbolic equilibria. More generally, twodifferent manifolds of equilibria cross each other, and at the crossing point theequilibria exchange their stability properties. However, beyond the bifurcationpoint, the number of equilibria does not change.


x1 = µx1 − x21

x2 = −x2.(8.73)

The equilibrium points are x(1) = (0, 0) and x(2) = (µ, 0). The eigenvalues of theJacobian at x(1) are λ1,2 = −1, µ, and at x(2) they are λ1,2 = −1,−µ. This


-3

-2

-1

0

1

2

3

X

-2 -1 0 1 2mu

Figure 8.22: Transcritical bifurcation diagram of (8.73)

−5 −4 −3 −2 −1 0 1 2 3−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1µ = 0

Figure 8.23: Phase portrait of (8.73) at bifurcation value.

means that for µ < 0, x(1) is a stable node and x(2) is a saddle point, and that forµ > 0, x(1) is stable and x(2) is saddle. An interchange of stability has ocurred,and we say a transcritical bifurcation has taken place at µ = 0. See bifurcationdiagram in Figure 8.22. In Figure 8.23 we show the phase portrait around (0, 0),which is a (nonhyperbolic) unstable point for µ = 0. Compare this with Figure8.24 where as the parameter µ goes from negative to positive, the equilibriumpoints interchange stability, e.g. the origin (0, 0) is stable for µ = −0.5 but itis a saddle point for µ = 0.5; at the same time, the equilibrium (µ, 0) is asaddle point for µ = −0.5, but it is a stable node for µ = 0.5. The system is notstructurally stable.

If we think of a macroeconomic model, one has to be extremely careful whenperforming inference; if a confidence region around parameter estimates includesa bifurcation point (x0, λ0), then various kinds of dynamics could be consideredconsistent, yet they may be in complete disagreement with the real problem.


−1 −0.5 0 0.5 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1µ = 0.5

−1 −0.5 0 0.5 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1µ = − 0.5

Figure 8.24: Phase portrait of (8.73): transcritical bifurcation.

-3

-2

-1

0

1

2

3

X

-0.5 0 0.5 1 1.5 2 2.5 3mu

Figure 8.25: Saddle-node bifurcation diagram of (8.74)

Although the illustration in Example 8.2.9 we have used the values µ = −0.5and µ = 0.5, this phenomenon is equally observed for arbitrarily small variationsaway from µ = 0. Hence, making inferences in real-world problems in such acase is a very delicate issue.

Saddle-Node bifurcations.

In this type of bifurcations, equilibria coalesce in such a way that the numberof equilibria can go from two to one to none: equilibria collide and annihilateone another. At the same time, stability properties change as the parameterpasses through the bifurcation value. These bifurcations are also known as foldbifurcations.



x1 = µ− x21

x2 = −x2.(8.74)

The equilibrium points are x(1) = (√µ, 0) and x(2) = (−√

µ, 0). The eigenvalues

−4 −3 −2 −1 0 1 2 3−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1µ = 0


of the Jacobian at x(1) are λ1,2 = −1, −2√µ, and at x(2) they are λ1,2 =

−1, 2√µ. The first thing we notice is that there are no equilibria for µ < 0,

−1 0 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1µ = − 0.25

−1 0 1−1.5

−1

−0.5

0

0.5

1

1.5µ = 0.5

Figure 8.27: Phase portrait of (8.74): saddle-node bifurcation.

there is only one equilibrium point for µ = 0 and two for µ > 0. The system isnot structurally stable. For µ > 0, x(1) is a stable node and x(2) is a saddle point.A saddle-node bifurcation has taken place at µ = 0. See bifurcation diagram inFigure 8.25. In Figure 8.26 we show the phase portrait around (0, 0), which isa (nonhyperbolic) unstable point for µ = 0. In Figure 8.27 for µ negative there


-1

-0.5

0

0.5

1

X

-0.4 -0.2 0 0.2 0.4 0.6 0.8mu

Figure 8.28: Pitchfork bifurcation diagram of (8.75)

are no equilibria; for µ positive, the equilibrium point (√µ, 0) is a stable node

and the equilibrium (−√µ, 0) is a saddle point.

Pitchfork bifurcations.

The main characteristic of a pitchfork bifurcation is that one equilibrium pointbifurcates into three equilibria, one of them changing the stability propertyand the other two keeping stability behavior. The bifurcation happens at anonhyperbolic equilibrium point and the system is not structurally stable.


x1 = µx1 − x31

x2 = −x2.(8.75)

The equilibrium points are x(1) = (0, 0), x(2) = (√µ, 0) and x(3) = (−√

µ, 0). We

observe that x(1) is the only equilibria for µ ≤ 0, and there are three equilibriafor µ > 0. Thus, a pitchfork bifurcation happens at µ = 0, where one equilibriumbifurcates into three as the parameter µ increases. See bifurcation diagram inFigure 8.28. The eigenvalues of the Jacobian at x(1) are µ and −1; at x(2) andat x(3) they are λ1,2 = −1, −2µ. Thus, for µ < 0, x(1) is a stable node; for µ >0, x(1) is a saddle, while x(2) and x(3) are stable nodes. For µ = 0, x(1) = (0, 0)is a (nonhyperbolic) stable equilibrium point. See behavior of solutions aroundit in Figure 8.29. Something similar happens for µ negative as we can see inFigure 8.30, where we also show solutions for µ positive. Observe how x(1) hasnow turned into a saddle point, and we go from one to three equilibrium points.


−1.5 −1 −0.5 0 0.5 1 1.5−1.5

−1

−0.5

0

0.5

1

1.5µ = 0


−1 0 1−1.5

−1

−0.5

0

0.5

1

1.5µ = −0.25

−1 −0.5 0 0.5 1

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

µ = 0.25

Figure 8.30: Phase portrait of (8.75): pitchfork bifurcation.

We insist on the fact that even though the illustration above was performed forµ±0.25, this radical change in the qualitative behavior of solutions happens forvalues of the parameter µ arbitrarily close to zero. This shows that in real-worldapplications it becomes crucial to know on which side of the origin the systemis operating, even when the system’s dynamics are observed to be stable.

Hopf bifurcations.

The bifurcations presented so far can be found even in a one-dimensional set-ting. However, for a Hopf bifurcation, we need the system to be at least two-dimensional. This type of bifurcation is very special because a periodic orbit isborn from an equilibrium point as the parameter varies, when such equilibriumchanges its stability properties. The amplitude of the periodic orbit at birth iszero, and it increases with the parameter. We show a Hopf bifurcation diagram


−1 −0.5 0 0.5 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1µ = 0


later in Section 8.3.


x1 = x1(µ− x21 − x2

2) − x2

x2 = x2(µ− x21 − x2

2) + x1.(8.76)

For any parameter µ, the only equilibrium point is (x1, x2) = (0, 0), and similarlyto Example 8.2.6, the system has a stable periodic orbit µ (cos t, sin t), for µ > 0.The Jacobian at (0, 0) has eigenvalues λ1,2 = µ± i, which implies that (0, 0) isa stable focus for µ < 0, and it is an unstable focus of µ > 0. Thus, theequilibirum (0, 0) changes from stable to unstable as the parameter goes fromnegative to positive, and at µ = 0 a stable periodic orbit bifurcates from (0, 0).This is what characterizes a Hopf bifurcation.

Also observe in Figures 8.31 and 8.32 that for µ ≤ 0 solutions spiral towards thestable equilibrium (0, 0), although for µ = 0, solutions first go around the origindescribing some kind of smaller and smaller circles, and for µ < 0 solutionsspiral towards the origin almost directly. On the other hand, for µ > 0, solutionsare attracted towards the stable periodic orbit, form outside and inside.

Note: The origin in Example 8.2.12 is nonhyperbolic and could be either astable focus or a center for µ = 0. Also recall that the Hartman-Grobmantheorem cannot be applied to this equilibrium point. By changing the system(8.76) to polar coordinates one can show that the origin is a stable focus forµ = 0.


−1 −0.5 0 0.5 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1µ = −0.5

−1 0 1−1.5

−1

−0.5

0

0.5

1

1.5µ = 0.5

Figure 8.32: Phase portrait of (8.76): Hopf bifurcation

Singularity bifurcations.

This type of bifurcations is less known than the ones introduced above, but theyare a very important phenomenon in applications, in particular, macroeconomicmodels [28, 29], and they are also very illustrative of the drastic changes inqualitative behavior that solutions show at a bifurcation value. The systemsinvolved are related to differential algebraic equations, and are a generalizationof the systems considered in the bifurcations above, namely

B(µ) x = f(x, µ). (8.77)

Thus, all cases considered before are a particular case with B(µ) = I. In thisnew setting, the matrix B(µ) may be singular, but its singularity itself does notnecessarily imply the presence of a singularity-induced bifurcation, which occurswhen the rank of B(µ) changes as the parameter µ varies.

What makes this type of bifurcation different from the cases considered beforeis the drastic change in the dimension of the phase portrait as the parametermoves through a bifurcation value. This is mainly a consequence of moving froma differential to a differential-algebraic system as the parameter varies, so thatthe algebraic equation imposes a restriction on the paths to be followed by thesolutions.

Example 8.2.13 We consider the system

x1 = x1(1 − x1)µ x2 = x1 − x2

2.(8.78)


−0.5 0 0.5 1 1.5−1.5

−1

−0.5

0

0.5

1

1.5µ = 2.0

−0.5 0 0.5 1 1.5 2−1.5

−1

−0.5

0

0.5

1

1.5µ = 0


This corresponds to B(µ) =

[

1 00 µ

]

in (8.77). There are three equilibria: x(1) =

(0, 0), x(2) = (1, 1) and x(3) = (1,−1). For any µ, (0, 0) is unstable; for µ <0, (1, 1) is unstable and (1,−1) is stable, while for µ > 0, (1, 1) is stable and(1,−1) is unstable. Thus, x(2) and x(3) interchange stability as µ crosses thevalue zero. At this bifurcation value µ = 0, when the rank of B(µ) decreasesfrom two to one, something more remarkable happens: the second equation in(8.78) becomes an algebraic constraint, forcing the the behavior of the systemto degenerate into motion only along the parabola x1 = x2

2. This results into adramatic drop in the dimension of the dynamics from a two-dimensional statespace to a one-dimensional curve. This is illustrated in Figure 8.33. For anegative value of µ, the solutions are somehow similar to those for positive µ,except of course that the stability properties of x(2) and x(3) are interchanged.

In practical applications, the potential presence of a singularity bifurcation sig-nals important implications for robustness of dynamic inferences, not only be-cause of the change of stability properties on both sides of the bifurcation values,but also because the dynamics of the system may drop into a ”black hole” lowerdimensional state space. Thus, in these cases a very high precision of parame-ter estimates is required, as the dynamics can be dramatically different withindifferent subsets of the parameter estimates’ confidence region.

8.3. PREDATOR-PREY MODELS WITH HARVESTING 449

8.3 Predator-Prey Models with Harvesting

To illustrate several of the concepts on dynamical systems introduced in thischapter we study some predator-prey models with harvesting. Although themodels introduced here are only two-dimensional systems, their dynamics arevery rich as they exhibit several types of equilibria, periodic orbits, bifurcations,connecting orbits, etc. On the other hand, we remark that predator-prey modelsin general are an active and important area of research in applied mathematics,and they play a central role in areas like biology, ecology and economics.

The inclusion of a harvesting component in a predator-prey system brings newinteresting mathematical results and its importance in resource managementcannot be overstressed. Researchers study the exploitation of resources includ-ing fish stocks and other species, and governments use the results to enact policywith the intention of avoiding overexploitation of certain species or the degra-dation of particular land areas, while commercial harvesting companies adjusttheir actions to maximize profits.

Let x(t) and y(t) denote the prey and predator populations respectively at timet. The following is a generalization of the classical Lotka-Volterra predator-preyrescaled model

x = x(1 − x) − axy1+mx

y = y(

−d+ bx1+mx

)

−H(y),(8.79)

where H is a function that defines harvesting, in this case on the predator.The parameters a, b, d and m are all positive. The parameter a represents thecapture rate of the prey, m represents the half-saturation constant, d is thenatural death rate of the predator and b represents the prey conversion rate.

The simplest case of harvesting is to assume that the function H is constant,that is, H(y) = h, for some positive constant h. A little more realistic as-sumption is to take H(y) = cy, for some positive constant c, that is, as thepredator population increases, so does its harvesting, to keep the population incheck. Mathematically, these two harvesting functions provide the system withinteresting dynamics; however they are not satisfactory from the point of viewof practical applications. By instance, as more predator become available, har-vesting more at a linear rate might not be profitable and eventually impossiblebecause of constrained resources for catching the predator.


0 1 2 3 4 50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

T

Figure 8.34: Harvesting function H(y)

Instead, here we consider the so called (continuous) threshold policy harvesting:

H(y) =

{

0 if y < Th(y−T )h+(y−T ) if y ≥ T,

(8.80)

where T is the threshold population size that determines when harvesting startsor stops and h is the rate-of-harvesting limit. In simple words, when a populationis above a certain level or threshold, harvesting occurs; when the population fallsbelow that level, harvesting stops. See Figure 8.34

For simplicity, H(y) in (8.80) can also be defined as a piecewise constant func-tion, jumping from no harvesting to some positive (relatively large) harvestingvalues. But that type of harvesting is impractical in real world application be-cause it would be difficult for managers to immediately harvest at a certain rateonce y ≥ T because of time delays and capital constraints. Instead, the con-tinuous harvesting (8.80) allows managers to smoothly increase the harvestingrate as the population increases. See Figure 8.34.

8.3.1 Boundedness of solutions

We are mainly interested in analyzing only the solutions in the closed first quad-rant R

2+ because any solutions outside of the first quadrant are not biologically

interpretable. Our first step into the analysis is to ensure that solutions startingon this first quadrant stay bounded. Accordingly, we have the following


Theorem 8.30 All solutions of the system (8.79) starting in the positive firstquadrant R

2+ are uniformly bounded.

Proof. Let w = x + aby. Then, w = x(1 − x) − ad

by − a

bH(y). For each k > 0

we havew + kw = x(1 − x+ k) + a

(

k−db

)

y − abH(y)

≤ (1+k)2

4 + a(

k−db

)

y − abH(y).

Choose k < d. Since 0 ≤ abH(y) ≤ ah

bfor y ≥ 0, there exists B > 0 such that

w+kw ≤ B, or w ≤ B−kw. Now consider the differential equation r = B−kw,with r(0) = w(0) = w0, whose solution r(t) = B

k(1− e−kt) +w0e

−kt is boundedfor t ≥ 0. Using a differential inequality [24] we get 0 < w(t) ≤ r(t) ≤ B

k, as

t→ ∞. That is, solutions stay in

S = {(x, y) ∈ R2+ : x+

a

by =

B

k+ γ, with γ > 0}.

�

8.3.2 Equilibrium point analysis

We find the equilibrium points of (8.79) and analyze their properties, by firsttaking m = b and by setting the growth rate of the prey to a. The followingtwo equilibria always exist:

P1 = (x1, y1) = (0, 0) (8.81)

andP2 = (x2, y2) = (1, 0). (8.82)

When y < T , we also have the point

P3 = (x3, y3) =

(

d

b(1 − d),b(1 − d) − d

b(1 − d)2

)

. (8.83)

This equilibrium point P3 ∈ R2+ when d < b

b+1 , which means that the predatorgrowth parameter b must be sufficiently large relative to the predator deathparameter d for there to be a coexistence equilibrium.

If we denote

Ny =

{

(x, y) : x =dy2 + (dh− dT + h)y − hT

b[(1 − d)y2 + (dT − dh− T )y + hT ]

}


and

Nx = { (x, y) : y = (1 − x)(1 + bx)},then, for y ≥ T , P+ = (x4, y4) are equilibrium points of (8.79), if they are inthe set Nx ∩Ny. These points P+ ∈ R

2+ when x4 ∈ (0, 1).

Next, we want to find out the stability properties of the equilibria. The Jacobianof (8.79) is

J(x, y) =

[

a[

1 − 2x− y(1+bx)2

]

− ax1+bx

by(1+bx)2

bx1+bx − d− ψ

]

, (8.84)

where ψ is given by

ψ =

{

0 if y ≤ Th2

(h+y−T )2if y > T.

The trace (T ) and determinant (D) of (8.84) are then given by

T = a

[

1 − 2x− y

(1 + 2x)2

]

+bx

1 + bx− d (8.85)

and

D = a

[

1 − 2x− y

(1 + bx)2

](

bx

1 + bx− d

)

+abxy

(1 + bx)3. (8.86)

We evaluate these two quantities at the equilibrium points and use the trace-determinant Theorem 8.5 to obtain:

a) At the point (x1, y1), the trace and determinant are

T = a− d and D = −ad < 0. Therefore, P1 is always a saddle point.

This tells us that it is rarely the case that the predator and the prey simultane-ously go extinct. Only when the initial condition starts along the stable branchof the origin does this happen.

b) At the point (x2, y2), the trace and determinant are

T = b1+b −a− d and D = a

[

d− b1+b

]

. Therefore, P2 has the following stability

conditions:

1. If b1+b > d, then (x2, y2) is a saddle point.


2. If b1+b < d and

(

b1+b − a− d

)2> 4a

(

d− b1+b

)

, then (x2, y2) is a stable

node.

3. If b1+b < d and

(

b1+b − a− d

)2> 4a

(

d− b1+b

)

, then (x2, y2) is a stable

spiral.

This means that if the death rate of the predator is sufficiently large, then theprey axis fixed point is stable. If the predator conversion rate b is sufficientlylarge however, then the prey axis fixed point becomes unstable as a saddle.Notice that no matter how large a value of b, if d > 1, then the predator cannotreproduce quickly enough to survive and the prey survives alone.

c) At the point (x3, y3), the trace and determinant are

T = ad[1+b(d−1)+d]b(d−1) and D = a

[

d− (1+b)d2

b

]

. Therefore, P3 has the following

stability conditions:

1. If b = 1+d1−d , then (x3, y3) is a center.

2. If d < b1+b and b < 1+d

1−d , then (x3, y3) is either a stable focus or a stablenode.

3. If d < b1+b and b > 1+d

1−d , then (x3, y3) is either an unstable focus or anunstable node.

Notice that the condition P3 ∈ R2++ implies d < b

b+1 , and on the other hand for

P3 to be a saddle, we need D < 0, or d > bb+1 . Thus, the coexistence equilibrium

P3 cannot be a saddle when it is biologically feasible.

(d) At the point (x4, y4), the trace and determinant are

T = −d− x(b+aλ)1+bx + ϕ and D =

ax[b−bx+λ(1+bx)(−d+ bx1+bx

+ϕ)]

(1+bx)2,

where λ = −1 + b− 2bx and ϕ = h2

[T−h+(x−1)(1+bx)]2.

As we can see, the mathematical expressions for the trace and determinant of theJacobian at P+ = (x4, y4) are more complicated, and in the expressions above, xis the solution of a polynomial equation of fifth degree with coefficients depend-ing on the parameters. Thus, the inequalities obtained from trace-determinantanalysis are much more involved, and we do not show them here, although theycan be explicitly found. See [39] for details. Numerically, however, we can stillobtain conclusions about the stability behavior for all the equilibria.


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.5

1

1.5

2

2.5

3

x

y

P3

(a) Unharvested.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.5

1

1.5

2

2.5

3

x

y

P+

(b) Harvested.

Figure 8.35: P3 = (0.05, 1.1875), P+ = (0.532, 1.65)

Numerical computations. To better understand the impact of continuousthreshold policy (CTP) harvesting on a predator-prey ecosystem, we rely onnumerical computations. While analytical derivations have helped to shed somelight on the properties of CTP, computational simulation can further determinewhether CTP can improve the conditions of the ecosystem.

The first case involves a high predation conversion rate parameter value b. With-out harvesting, the coexistence equilibrium is an unstable focus. The harvestingagent can use CTP to stabilize the system by choosing a sufficiently high harvest-ing limit h. See Figure 8.35. Notice that in this case, CTP increases the long-term populations of both the predator and prey while simultaneously stabilizingthe populations. The parameters used are a = 0.5, b = 5, d = 0.2, h = 4, T = 0.5.

The second case is evidence for the notion that CTP cannot damage the ecosys-tem when it is in a stable state, i.e. enacting CTP does not cause the coexistenceequilibrium to become unstable. Suppose harvesting agent enacts CTP butchooses large parameter values for h and T to determine if CTP can negativelyaffect the coexistence equilibrium. See Figure 8.36. The predator is now beingharvested under extreme effort with a low threshold (T ) and a high harvestinglimit h. Nevertheless, the coexistence equilibrium is still stable. We have usedthe parameter values a = 0.25, b = 0.5, d = 0.3, h = 10, T = 0.05.

8.3.3 Bifurcations

As remarked before, given a dynamical system, one of the central questions ishow the given model depends on the choice of the parameters, or more precisely,


0 0.2 0.4 0.6 0.8 1 1.2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

y

P3

(a) Unharvested.

0 0.2 0.4 0.6 0.8 1 1.2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

y

P+

(b) Harvested.

Figure 8.36: P3 = (0.8571, 0.2041), P+ = (0.9654, 0.0513)

how the qualitative behavior of solutions changes as one or more parameters areallowed to vary. In this section we track for possible bifurcations as they rep-resent drastic changes in the predator-prey system. Our model (8.79) exhibitsseveral bifurcations including fold and Hopf bifurcations when we allow param-eters to vary continuously.

In Figure 8.37 (a) we show the basic bifurcation diagram, when the parameterh varies continuously. Two fold bifurcations are detected at h ≈ 0.4055193and h ≈ 0.7701848. At these values of the parameter h, solutions change theirproperties from stable to unstable and back to stable. This bifurcation diagramalso allows to see the number of equilibria: e.g. the system has three equilibriafor 0.4055194 ≤ h ≤ 0.7701848, two of them stable and one unstable.

Starting at the fold bifurcation for h ≈ 0.7701848, we have can perform two-parameter continuation by allowing both h and m to vary at the same time.This gives the bifurcation diagram shown in Figure 8.37 (b), which representsa curve of fold points; this means that along this curve in the (h,m) space, thesystem (8.79) has a fold.

If instead of h we set m as the main parameter of continuation, then a Hopfbifurcation is detected. Indeed, for m ≈ 1.7 such bifurcation is numericallydetected, where the equilibrium (x2, y2) is a center. From here, a branch ofstable periodic orbits is born. See bifurcation diagram in Figure 8.38.

8.3.4 Connecting orbits

We can see from Figure 8.38 that periodic solutions are born at an equilibriumpoint as the parameter passes through the Hopf bifurcation value. Here, we first


0

0.2

0.4

0.6

0.8

1

1.2

1.4

X

0 0.2 0.4 0.6 0.8 1h

(a) Basic bifurcation diagram

0

1

2

3

4

5

6

7

8

m

0 0.2 0.4 0.6 0.8 1h

(b) Two-parameter bifurcation diagram

Figure 8.37: Bifurcation diagrams of predator-prey model

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

X

0.5 1 1.5 2 2.5m

Figure 8.38: Hopf Bifurcation gives out periodic solutions


0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.553.5

4

4.5

5

5.5

6

6.5

7

7.5

h = 0h = 0.06h = 0.1h = 0.164h = 0.167

Figure 8.39: A branch of periodic orbits of (8.79)

explicitly compute one such periodic solution, and then we allow the parameterh to vary to compute a branch of stable periodic orbits. Later we computeheteroclinic connections, more precisely, point-to-periodic connections. We startwith h = 0.1 and we follow the periodic orbits down to smaller values of h downto h = 0 (no harvesting), and also to larger values of h until the correspondingstable Floquet multiplier approaches the value 1, and the periodic orbit collapsesto a point. See Figure 8.39 and Table 8.2.

h Period Floquet multiplier

.0 29.3147399 .3303902374914.08 28.2861845 .4676316116910.13 27.6349319 .6953439617105.16 27.2112798 .9301526888350.167 27.1082952 .9996358112280

Table 8.2: Period and Floquet multipliers

Each periodic solution shown in Figure 8.39 represents long-term coexistenceof predator and prey for different harvesting efforts h. Observe that for largervalues of h we have relatively smaller values of both populations, though coex-istence is preserved.

Clearly, if an initial value for the system (8.79) is on one of its periodic orbits,


0 0.5 10

2

4

6

8h = 0.164

0 0.5 10

2

4

6

8h = 0.1

0 0.5 10

2

4

6

8h = 0

Figure 8.40: Branch of point-to-periodic connections of (8.79)

then long-term coexistence is guaranteed. The question is on how one can arriveto such solutions from a given equilibrium point. Point-to-periodic solutionsrepresent in this case paths that explicitly describe the evolution of predator andprey populations from say, an unstable state, to a (long-term) stable one. Wecompute such heteroclinic orbits allowing the parameter h to vary and computea branch of point-to-periodic connections. Given an equilibrium point, we startin the direction of its unstable manifold to arrive through the stable manifoldof the periodic orbit. This tells us how from an equilibrium point of the system(8.79) one can manipulate or perturb the predator and prey populations to forcethe system to arrive to a long-term stable periodic solution. See Figure 8.40.

8.3.5 Other models

A long list of other predator-prey models can be studied, with different inter-actions between predator and prey and with different harvesting functions. Byinstance, one can study the so-called ratio-dependent models, whereby the rateof prey consumption does not just depend on the prey population, but insteadon the ratio of predator and prey population. In such models the interaction of


predator and prey is represented as

axy

my + x.

One can also consider higher-dimensional systems to study the cases of say twoor more prey and one predator, or two or more predator and one prey. Higher-dimensional systems are also obtained when considering a disease acting on oneor both species, as one has to add equations to consider infected populations.

If one considers the motion of one or more of the species, then diffusion termshave to be added to the model, leading to a system of partial differential equa-tions. This kind of systems can be transformed into a classical system of ordinarydifferential equations by certain change of coordinates. This is done for instancewhen studying the corresponding traveling wave solutions of the system.

Finally, one can consider different harvesting functions, e.g. piecewise continu-ous functions, or periodic functions to represent seasonal or rotational harvest-ing, and harvesting could be applied to either or both, predator and prey.

For illustration, one can slightly modify the original model (8.79) to the followingone

x = x(1 − x) − axy1+mx −H(x)

y = y(

−d+ bx1+mx

)

,(8.87)

where now the harvesting is on the prey and it is given by the following piecewiselinear function (that follows about the same shape of that in (8.80) ).

H(x) =

0 if x < T1h(x−T1)T2−T1

if T1 ≤ x < T2

h if x ≥ T2.

(8.88)

Here, we have two threshold values, T1 and T2, and one has to study separatelythe equilibrium points for the three regions: x < T1, T1 ≤ x < T2, and x ≥T2. Then, one can follow a similar study to the one introduced above, to findpossible periodic solutions and their stability properties, and one can also trackfor bifurcations and special solutions.

We have computed two bifurcations that are clearly shown in Figure 8.41. Thereis a transcritical bifurcation at h ≈ 2.2, where two equilibria interchange stabil-ity, and a fold bifurcation at h ≈ 2.5, where one stable equilibria collides with


-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

X

-0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4h

Figure 8.41: Fold and transcritical bifurcations of (8.87), (8.88)

an unstable one. Figure 8.42 shows a Hopf bifurcation, giving out a branch ofperiodic orbits. These bifurcation diagrams of model (8.87) already show theexistence of several interesting solutions with different stability properties andbifurcations. In general a rich dynamics of the system should be found. Weleave most computations and analysis of this and other models as exercises forthe reader.

Figure 8.42: Hopf Bifurcation of (8.87), (8.88)

8.4. FINAL REMARKS AND FURTHER READING 461

8.4 Final Remarks and Further Reading

We have presented a short introduction to some basic topics on dynamical sys-tems. We started with linear systems x = Ax and the study of their solutionsthrough matrix exponentials and similarity. The concept of stability and theasymptotic behavior of solutions as well as their corresponding invariant sub-spaces were studied in some detail, using two-dimensional systems as an intro-ductory step.

We have studied nonlinear dynamical systems x = f(x) locally and globally,mostly from the theoretical point of view, but we have taken care of introducingsome numerical techniques to compute some special solutions. Recall that ingeneral it is not possible to have an explicit or analytical solution of a nonlineardynamical system, but the theory introduced allows us to study the qualitativebehavior of solutions, without explicitly knowing them. At this point, somenumerical software becomes essential to help understand the general behaviorof solutions.

Linearization has been the central tool to study nonlinear systems locally, thanksespecially to two main theorems: the stable manifold theorem and Hartman-Grobman theorem, which in both cases, take hyperbolicity as the main assump-tion. We then have briefly studied bifurcations, chaos and connecting orbits, inan effort to understand a nonlinear system beyond their local behavior.

Although we have limited ourselves to continuous dynamical systems, a similarstudy can be done for discrete systems, as well as for systems with delay andother special properties. There is a long list of excellent references to studythese and other very important areas of dynamical systems in great detail, fromboth, the theoretical and numerical point of view. For an excellent introductioninto these topics, the books by Perko [46] and Wiggins [57] are a great start.For a little more advanced study, besides the articles cited within this chapter,see the books by Guckenheimer and Holmes [23], and Chow et al. [12]. A greattheoretical as well as numerical study of dynamical systems can be found in thebooks by Dieci and Eirola [14], Hale and Kocak [25], and Kuznetsov [36].

8.5 Exercises


Exercise 8.1 True or False? The origin x = 0 is the only equilibrium point ofa given linear system x = Ax.

Exercise 8.2 Let A be an n× n matrix. Show that ‖eA‖ ≤ e‖A‖.

Exercise 8.3 Let A be an n× n matrix and let τ > 0. Show that the series

∞∑

k=0

Aktk

k!

is uniformly and absolutely convergent for any |t| ≤ τ .

Exercise 8.4 Let v be an eigenvector of a matrix A corresponding to an eigen-value λ. Show that v is also an eigenvector of eA, with eigenvalue eλ.

Exercise 8.5 Let λ1, . . . , λn be the eigenvalues of a matrix An×n, and let a realnumber γ satisfy γ > max

j=1,...,nRe(λj). Show that

‖eAt‖ ≤ Ceγt,

for some C > 0.

Hint: Use Exercise 7.52.

Exercise 8.6 Assume that all eigenvalues of a matrix An×n have negative realparts. Show that there exist positive constants K and α such that every solutionx(t) of x = Ax satisfies

‖x(t)‖ ≤ Ke−αt, t ≥ 0.

That is, all solutions approach zero as t → ∞.


Exercise 8.7 Consider the system x = Ax+ f(t), and assume that there existconstants α, C > 0, t0 ≥ 0, such that |f(t)| ≤ Ceαt, for t ≥ t0. Show that everysolution x(t) of the system satisfies

|x(t)| ≤ Keβt, t ≥ t0,

for some constants β and K > 0.


8.5. EXERCISES 463

Exercise 8.8 The following matrices have the same eigenvectors. Sketch thephase portrait of the corresponding system x = Ax.

(a) A =

[

1 20 3

]

, (b) A =

[

3 −20 1

]

, (c) A =

[

−3 20 −1

]

(d) A =

[

−1 40 3

]

, (e) A =

[

3 −40 −1

]

, (f) A =

[

0 −30 −3

]

Exercise 8.9 Let A and B be two matrices such that AB = BA. Show thateA+B = eAeB.

Exercise 8.10 Let A =

[

λ 10 λ

]

. Show that

eAt = eλt[

1 t0 1

]

.

Hint: Let B =

[

0 10 0

]

, write A = λI +B and use Exercise 8.9.

Exercise 8.11 Let A =

[

a −bb a

]

. Show that

eAt = eat[

cos bt − sin btsin bt cos bt

]

.

Hint: First use induction to show that Ak =

[

Re(λk) −Im(λk)Im(λk) Re(λk)

]

, where

λ = a+ ib.

Exercise 8.12 Let u(t) and v(t) be solutions of the system x = Ax. Show thatany linear combination of u(t) and v(t) is also a solution of the system.

Exercise 8.13 For the following matrices

(a) A =

[

3 41 3

]

, (b) A =

[

5 43 1

]

,

(c) A =

[

2 −24 6

]

, (d) A =

[

0 4−1 0

]

solve the corresponding linear system x = Ax, and sketch the phase portraits onboth, the (x1, x2) and the (y1, y2) planes, where as usual y = P−1x. Indicatewhether the origin is stable, unstable, a node, a focus or a center.


Exercise 8.14 Solve the IVP x = Ax, x(0) = [c1 c2]T , and sketch the phase

portraits corresponding to the matrices

A =

[

−3 00 0

]

, and A =

[

0 00 0

]

.

Indicate whether the origin is stable, unstable, a node, a focus or a center.

Exercise 8.15 Consider the system

x1 = ax1 − x2

x2 = x1 + bx2.

Under which conditions on the parameters a and b is the origin a saddle, acenter or a stable/unstable focus?

Exercise 8.16 Let A be an n× n matrix and λ an eigenvalue of A. We definethe generalized eigenspace of A corresponding to λ as the set Eλ of all generalizedeigenvectors of A associated to λ, together with the zero vector. Show that Eλis invariant under A, that is, AEλ ⊂ Eλ.

Exercise 8.17 Let A be an n × n matrix. Show that the subspaces Es, Ec, Eu

are invariant with respect to the system x = Ax, and that

Rn = Es ⊕ Ec ⊕ Eu.

Hint: For the first part, use Exercise 8.16.

Exercise 8.18 Consider the system x = Ax, where A =

−4 0 −12 6 734 0 −4

. Find

the invariant subspaces Es, Ec, Eu of the system.

Exercise 8.19 Consider the system x = Ax, where A =

−1 −10 −100 10 0

10 −11 −1

.

Find the invariant subspaces Es, Ec, Eu of the system and then sketch the phaseportrait.

8.5. EXERCISES 465

Exercise 8.20 Find the subspace Ec and plot the solution curves of the system

x = Ax, where A =

[

0 30 0

]

.

Exercise 8.21 Find the subspaces Es, Ec, Eu and plot the solution curves ofthe system x = Ax, where

(a) A =

−1 −1 21 −1 00 0 3

, (b) A =

−2 1 20 5 10 5 1

.

Exercise 8.22 (Floquet’s theorem). Consider the linear system x = A(t)x,where A(t) is a continuous periodic matrix function of period τ , and let Φ(t)be any fundamental matrix solution of such system. Show that there exists aconstant matrix B and a nonsingular periodic matrix function P (t) of period τsuch that

Φ(t) = P (t) eBt.

Hint: Use the facts that Φ(t+ τ) is also a fundamental matrix solution of thesame system, so that Φ(t + τ) = Φ(t)C, for some constant matrix C, and thatthere exists a constant matrix B such that C = eBτ .

Exercise 8.23 Under the hypotheses of Exercise 8.22, show that x(t) is a so-lution of x = A(t)x if and only if y(t) = P−1(t)x(t) is a solution of y = By.

Exercise 8.24 Consider the nonhomogeneous system x = A(t)x+ f(t), with Aand f continuous and periodic, with the same period τ . Show that a solutionx(t) of such system is periodic of period τ if x(τ) = x(0).

Exercise 8.25 Consider the nonhomogeneous system x = A(t)x + f(t), with

A =

[

0 1−2 −4

]

and f(t) =

[

0cos t

]

. Show that the matrix e2πA − I is

invertible.

Exercise 8.26 For the matrix A and function f of Exercise 8.25, find a vectorz ∈ R

2 such that

z = e2πAv + e2πA∫ 2π

0e−sAf(s) ds.


Exercise 8.27 (Gronwall’s inequality.) Let u(t) be a nonnegative continu-ous function such that for positive constants K and C we have

u(t) ≤ K + C

∫ t

0u(s) ds,

for all t ∈ [0, a]. Show that

u(t) ≤ KeCt,

for all t ∈ [0, a].

Exercise 8.28 Let E ⊂ Rn be open and f : E → R

n. Show that if f ∈ C1(E),then f is locally Lipschitz on E.

Exercise 8.29 Let E = [0, 1] × [0, 1]. Show that the function f : E → R2

defined by f(x, y) = (x+ y, xy) is Lipschitz.

Exercise 8.30 Find the stable and unstable manifolds S and U respectively, ofthe system

x1 = 3x1

x2 = −x21 − x2

by first finding the solution explicitly.

Exercise 8.31 Find the stable and unstable manifolds S and U respectively ofthe system of Exercise 8.30 by using successive approximations.

Exercise 8.32 Let H = I − F , where F : Rn → R

n. Show that if F is acontraction, then H is a homeomorphism from R

n to Rn.

Exercise 8.33 Let a and b be positive real numbers and consider the systems

x = ax and x = bx.

Show that these two systems are equivalent by finding a homeomorphism h thatmaps trajectories of the first system onto trajectories of the second one.

8.5. EXERCISES 467


x1 = −2x1 + x22

x2 = −x1 + x+ 2.

Show that the system has two equilibria, and find the linearizations of this systemaround those points. Compute and plot the solutions of both, the linear andnonlinear systems.


x1 = 2x1 − x1x2 + 12x

21

x2 = 12x2 − x1x2.

Show that the origin is hyperbolic and find the linearization of this system aroundthe origin. Compute and plot the solutions of both, the linear and nonlinearsystems.

Exercise 8.36 Consider the following equation for a pendulum with damping

y′′ + 2ay′ + b2 sin y = 0.

Show that the origin is asymptotically stable for any a > 0, b > 0, by first trans-forming the equation into a two-dimensional system of first order x = f(x, a, b),and then linearizing around the origin.


x1 = x2

x2 = −ax21x2 − x1.

Find the linearization of this system around the origin, and show that the eigen-values of the corresponding matrix are purely imaginary for any a ∈ R. Plotthe solutions of the nonlinear system and verify that the origin is attracting fora > 0 and repelling for a < 0. Does this contradict Hartman-Grobman theorem?

Exercise 8.38 Consider the Lorenz system (8.64), with b = 1 but σ and λfree. Find all equilibrium points of the system. What stability properties has theorigin when λ < 1? Show that the other two nontrivial equilibria are sinks aslong as 1 < λ < σ(σ+4)

σ−2 .



x1 = µx1(x21 + x2

2) + x2

x2 = µx2(x21 + x2

2) − x1.

First, verify that the eigenvalues of the Jacobian of the system at the origin arepurely imaginary for all µ ∈ R. Show however that for µ < 0 the origin isasymptotically stable and for µ > 0 it is unstable.

Hint: Compute the derivative of the square of the distance of any solution(x1(t), x2(t)) to the origin.


x1 = x1(1 − x21 − x2

2) + x2

x2 = x2(1 − x21 − x2

2) − x1

Use polar coordinates (r, θ) to rewrite the system as

r = r(1 − r2)

θ = −1.

With r(0) = r0 and θ(0) = θ0, show that the solution is given by

r(t) =r0

√

r20 + (1 − r20)e−2t

, θ(t) = −t+ θ0

and that trajectories approach the unit circle as t→ ∞.

Exercise 8.41 Consider the following predator-prey model with diffusion

{

(u1)t = au1(c− u1) − u1u2

u1+1

(u2)t = (u2)xx − u2 + b u1u2

u1+1 .(8.89)

Introduce the traveling wave coordinates

u1(t, x) = v1(x+ st) = v1(z), u2(t, x) = v2(x+ st) = v2(z)

and v3 = v2, where the dot denotes derivative with respect to z, to convert(8.89) into a 3-dimensional system v = f(v). Compute a branch of periodicorbits (periodic traveling wave solutions) for a = 2, b = 4, c = 3 by setting thespeed wave s as a free parameter.

8.5. EXERCISES 469

Exercise 8.42 Show that Γ(t) = (0, cos t, sin t) is a periodic solution of

x1 = −x1x22 − x1x

23

x2 = x21x2 − x3

x3 = x2 + x21x3

and find the Floquet multipliers of the periodic orbit.

Exercise 8.43 Let D be a closed and bounded set in Rn and for f defined and

continuous on [0, τ ] ×D, for some τ > 0, consider the system

x = f(t, x)f(t, x) = f(t+ τ, x).

Assume that for sufficiently small ǫ > 0, x+ ǫf(t, x) ∈ D, for all t and all x onthe boundary of D. Show that the system has a periodic solution of period τ .


x = A(t)x+ f(t), a ≤ t ≤ bBax(a) +Bb x(b) = c,

(8.90)

with solution

x(t) = Φ(t)Q−1c+

∫ b

a

G(t, s)f(t) ds

satisfying ‖x‖∞ ≤ k1‖c‖∞ + k2‖f‖1, where

k1 = ‖Φ(t)Q−1‖∞, and k2 = supa≤t, s≤b

‖G(t, s)‖∞.

Show that k1, k2 and the function G are independent of the fundamental matrixsolution Φ(t).

Exercise 8.45 Consider the following perturbation of the system (8.90)

y = A(t)y + f(t) + g(t), a ≤ t ≤ bBay(a) +Bb y(b) = c+ d.

(8.91)

Define e(t) = x(t) − y(t) and show that ‖e‖∞ ≤ k1‖d‖ + k2‖g‖∞.

Hint: First consider the system satisfied by e(t) by combining (8.90) and (8.91).


Exercise 8.46 Consider the Kuramoto-Sivashinsky equation

ut + uxxxx + uxx + uux = 0,

which models spatio-temporal evolution of a flame front. Take the change ofvariables u(x, t) = v(x) − s2t, y = v′ to get

y′′′ + y′ +1

2y2 − s2 = 0.

Now, as usual, write this last equation as a system u = f(u). For the travelingwave speed s = 1 the system has two (hyperbolic) periodic orbits. Compute thoseand find the dimension of the corresponding stable and unstable manifolds.

Exercise 8.47 Prove Theorem 8.25.


x1 = 0.2x1 + x2

x2 = −x1 − x3

x3 = x2x3 − µx3 + 0.2.

For µ = 2.2, compute the solution starting at (0.6472, 3.7669, 1.9206) and youwill see a periodic orbit of period τ1 ≈ 5.765. For µ = 3.1, start the solutionat (−1.9341, 4.6617, 0.3610) and now you will see a periodic orbit of periodτ2 = 2τ1. This process repeats as µ increases. Such phenomenon is known asperiod doubling bifurcation.

Exercise 8.49 Consider the predator-prey system (8.87), (8.88). Find the equi-librium points and their stability properties. Then compute a branch of periodicorbits that appear at the Hopf bifurcation.

Exercise 8.50 Consider the following ratio-dependent predator-prey model withlinear harvesting on the prey

x = x(1 − x) − axyy+x − hx

y = y(

−d+ bxy+x

)

.(8.92)

(a) Find the equilibrium points of the system and determine the values of theparameters to guarantee that these points represent coexistence of both species

(b) Set a = 2, b = 1, d = 0.75, and for increasing values of h, starting ath = 0.3126, compute a branch of connecting orbits from the equilibrium (1−h, 0)to a branch of (stable) periodic orbits.

8.5. EXERCISES 471


u1 = u1 + βv1 − u31 − 3u1u

22 − u1(v

21 + v2

2) − 2v1v2u2

v1 = −βu1 + v1 − v31 − 3v1v

22 − v1(u

21 + u2

2) − 2u1u2v2u2 = (1 − 2λ)u2 + (β − 2λ)v2 − u3

2 − 3u21u2 − u2(v

21 + v2

2) − 2v1v2u1

v2 = −(β + 2λ)u2 + (1 − 2λ)v2 − v32 − 3v2

1v2 − v2(u21 + u2

2) − 2u1u2v1,

where β = 0.55 and λ is a free parameter. This system has the following periodicorbits

y−(t) = ( 0, 0, ρ(t) cos θ(t), ρ(t) sin θ(t) ), y+(t) = (cos βt, − sin βt, 0, 0),

whereρ = (1 − 2λ− 2λ sin 2θ)ρ− ρ3 , θ = −β − 2λ cos 2θ .

Compute a branch of connecting orbits from y− to y+ starting at λ = 0.1 andfind the dimension of the corresponding stable and unstable manifolds. To plotthe solutions in three dimensions, use the coordinates (u2, v2,

√

u21 + v2

1).

Exercise 8.52 Experiment the sensitive dependence of solutions on the initialconditions for the Lorenz system (8.69), with the same value parameters σ =10, λ = 28, b = 8/3, by first using the initial condition x(0) = (0, 0.01, 0) andthen x(0) = (−0.01, −0.01, 0). Plot and compare both solutions.


x1 = −x2 − x3

x2 = x1 + ax2

x3 = b+ x3(x1 − c).

Take a = 0.15, b = 0.2, c = 10 and compute the corresponding solution startingat x(0) = (10, 0, 0). You should obtain the so called Rossler attractor.

Exercise 8.54 Consider the one-dimensional dynamical system x = f(x, λ),with λ ∈ R, and assume that for some (x0, λ0) ∈ R×R the following conditionshold:

f(x0, λ0) = 0,∂f

∂x(x0, λ0) = 0,

∂2f

∂x2(x0, λ0) 6= 0,

∂f

∂λ(x0, λ0) 6= 0

Show that x = f(x, λ) undergoes a saddle-node bifurcation at λ = λ0.

Hint: Using the first and last conditions above, apply the implicit functiontheorem to show that there is a smooth curve λ = λ(x) of equilibria, then showthat its graph is concave.


Exercise 8.55 Consider the one-dimensional equation

x = f(x, a, b) = a+ bx− x3.

(a) Fix a = 0 and study the number of equilibria and their stability for b > 0and for b ≤ 0. Verify that this gives a pitchfork bifurcation.

(b) Solve simultaneously the equations f(x, a, b) = 0 and ∂f∂xf(x, a, b) = 0 to get

both a and b in terms of x and then combine the expressions for a and b to get27a2 = 4d3. Plot this on the (a, b) plane. The curve you obtain represents acusp bifurcation.

Exercise 8.56 Consider the predator-prey system

x1 = 2x1(1 − x1) − 2x1x2

µ+x1

x2 = −x2 + 2x1x2

µ+x1.

Show that (µ, 2µ(1 − µ) ) is an equilibrium point, and that for µ = 1/3, theJacobian at that equilibrium has purely imaginary eigenvalues. Then plot thesolutions for µ < 1/3, µ = 1/3 and µ > 1/3 to verify that a Hopf bifurcationoccurs.


x1 = x1(1 − x1)µx2 = x1 − x2.

Find the equilibrium points and their stability properties. Plot the phase portraitsfor µ = 0 and µ = 2 to see a drop on the dimensions. This is a singularitybifurcation.


x1 = −x1 sinµ− x2 cosµ+ (1 − x21 − x2

2)2(x1 cosµ− x2 sinµ)

x2 = x1 cosµ− x2 sinµ+ (1 − x21 − x2

2)2(x1 sinµ+ x2 cosµ).

Transform this system to polar coordinates and verify that for µ < 0 thereare no periodic orbits because r > 0, that for µ = 0, the unit circle is anunstable (saddle) periodic orbit, and that for sufficiently small µ > 0, there aretwo periodic orbits, one stable, with radius less than 1, and one unstable, withradius greater than 1. Plot the solutions for these three cases. This phenomenonis known as saddle-node bifurcation of periodic orbits.

8.5. EXERCISES 473


x1 = x2

x2 = x1 − x21 + µx2.

Compute and plot the solutions for µ = −1, µ = 0 and µ = 1. You will seea so-called homoclinic bifurcation: for µ = 0 there is a homoclinic orbit to theorigin, and a set of periodic orbits around (1, 0), but the loop and the orbits arebroken for µ 6= 0.


x1 = 1 − x21 + µx1x2

x2 = x1x2 + µ(1 − x21)

The system has two equilibria, (−1, 0) and (1, 0). Are these equilibria hyperbolic?Compute and plot the solutions for µ = −1, µ = 0 and µ = 1 around theequilibrium points. A so called heteroclinic bifurcation happens at µ = 0, whenthe two equilibria are connected.

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